What are BMS supertranslation symmetries?
The supertranslations are just direction-dependent time translations on the boundary of asymptotically flat spacetimes. Specifically, if the "future null infinity" boundary of such a spacetime is given retarded-time coordinate $u$, then a supertranslation transforms that coordinate according to \begin{equation} u \to u + \alpha(\theta, \phi), \end{equation} where $\alpha$ is any (differentiable) function of the angular coordinates $(\theta, \phi)$. The idea is actually pretty simple, though the explanation is a bit long. For simplicity, I'll just stick to the future in this answer, but all the same ideas apply on past null infinity if you replace the retarded time $u$ with advanced time $v$, and future-directed null cones with past-directed null cones.
In asymptotically flat spacetimes, the supertranslations are one of the asymptotic symmetries of the metric. Naively, you might expect just the Poincaré group (rotations, boosts, time translation, and space translations), since that describes all the symmetries of Minkowski spacetime. But it turns out that at the boundary of asymptotically flat spacetime, you get the Poincaré group, but you also get "generalized" translations. Putting those all together, you get the whole BMS group. The supertranslations include simple time translation, and space translations, as well as these "generalized" translations.
The first full exposition of the BMS group was by Sachs in this paper, which is worth reading if you want to understand the history of this this group. He called it the "generalized Bondi-Metzner group", because it wouldn't be appropriate to insert his own name; other people did that for him later, and it's now called the Bondi-Metzner-Sachs group.
Okay, now on to the explanation...
There's a pretty simple interpretation that lets you understand the supertranslations, starting with time translation, building to space translations, and finally ending up at general supertranslations. I think a really nice pedagogical explanation of this is given in section II.B of this paper (though I'm a little biased, because I wrote that paper). For simplicity, I'll just talk about the most basic example: Minkowski space. But any asymptotically flat spacetimes that we discuss look more-or-less like Minkowski asymptotically, and an even larger class of spacetimes are close enough for this example to help understand.
To start off with, we have to consider the entire Minkowski spacetime along with its actual asymptotic limits, which is done with the "compactified" spacetime. Basically, we just draw a Penrose diagram, which changes coordinates so that we can draw infinitely distant points on a finite diagram. The interesting part of this diagram for our purposes is future null infinity, $\mathscr{I}^+$, which is the (future) asymptotic limit of where null signals go to in an asymptotically flat spacetime. The Minkowski spacetime is a nice model for explaining this because we can construct coordinates on $\mathscr{I}^+$ by shooting light rays from inertial emitters inside the spacetime. (More complicated spacetimes might have black holes or other complications that would make it impossible to define coordinates covering $\mathscr{I}^+$ in this way, but don't change the essential features we care about — the asymptotic structure.)
Suppose an inertial observer $\mathscr{A}$ has the usual spherical coordinates $(\theta, \phi)$, and has some clock for their proper time $\tau$. That observer shoots a light ray in the direction $(\theta, \phi)$ at time $\tau$, and the light ray eventually approaches some point on $\mathscr{I}^+$. So now, we just label that point with the angular coordinates $(\theta, \phi)$ and a retarded time $u = \tau$. But maybe we decide that observer $\mathscr{A}$'s clock was wrong by some amount $\delta \tau$. The light ray that was emitted at $\tau$ should have been emitted at $\tau - \delta \tau$. I've copied Figure 3 from my paper, showing exactly this, below. If we make this adjustment, all that happens to the coordinates on $\mathscr{I}^+$ is that \begin{equation} \tag{1} u \to u - \delta \tau. \end{equation} That's the first type of supertranslation — a simple time translation. And it shouldn't surprise you that the asymptotic metric "on" $\mathscr{I}^+$ is not altered by the coordinate transformation caused by this time translation. This situation is illustrated in figure 3 from my paper, which is also shown below.
Now suppose we have another observer, $\mathscr{B}$, that is stationary with respect to $\mathscr{A}$, but is displaced by some $\delta \boldsymbol{x}$. This observer also gets to shoot off light rays and label $\mathscr{I}^+$. But as we see in figure 4 (below), the light rays that $\mathscr{B}$ shoots off in two opposite directions reach points on $\mathscr{I}^+$ that $\mathscr{A}$ could only reach by shooting photons at different times depending on the direction: $\mathscr{A}$ has to shoot off the one going left much earlier than the one going right. In fact, if their clocks are synchronized and $\mathscr{B}$ emitted both photons at $\tau=0$, then it's not hard to see that $\mathscr{A}$ had to emit to the left at $\tau=-\lvert \delta \boldsymbol{x} \rvert$ and to the right at $\tau=\lvert \delta \boldsymbol{x} \rvert$, to account for the extra propagation times. Of course, this diagram is simplified because it suppresses two dimensions of our four-dimensional spacetime. But it's easy to figure out that in any direction $\hat{\boldsymbol{r}}$, the retarded time on $\mathscr{I}^+$ is transformed by this spatial translation as \begin{equation} \tag{2} u \to u + \delta \boldsymbol{x} \cdot \hat{\boldsymbol{r}}. \end{equation} That's the second type of supertranslation — a simple space translation. And again it shouldn't surprise you that the asymptotic metric "on" $\mathscr{I}^+$ is not altered by the coordinate transformation caused by this space translation, since the Minkowski metric is not altered by it.
You might observe that in terms of the usual spherical-harmonic index $\ell$, transformation (1) above is an $\ell=0$ function of coordinates on the sphere (independent of direction), while transformation (2) is an $\ell=1$ function. That is, if we think of them as being expanded in spherical harmonics $Y_{\ell, m}$, then we can write those transformations as \begin{gather} \tag{1'} u \to u + \alpha^{0,0}Y_{0,0}(\theta, \phi), \\ \tag{2'} u \to u + \sum_{m=-1}^{1} \alpha^{1,m}Y_{1,m}(\theta, \phi), \end{gather} where we just have a simple condition on $\alpha^{0,0}$ and the $\alpha^{1, m}$ to ensure that the result is purely real. It is natural to wonder what a general transformation like \begin{gather} \tag{3} u \to u + \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell} \alpha^{\ell,m} Y_{\ell,m}(\theta, \phi) \end{gather} would mean. (Again, assuming the $\alpha^{\ell,m}$ obey the reality condition.) This is what we mean by a general supertranslation.
This generalization is interesting, but it wouldn't really mean much unless we had some physically relevant fact about it. It turns out that a supertranslation is also a symmetry of the asymptotic metric. Obviously, it's not generally a symmetry of the metric in the interior of the spacetime, but something important happens once we get to the boundary $\mathscr{I}^+$.
There's a pretty simple intuitive explanation of why supertranslations are relevant asymptotically. Very roughly speaking, "neighboring" points on $\mathscr{I}^+$ with infinitesimally different $(\theta, \phi)$ coordinates are actually infinitely "far apart" in space. More precisely, they are causally disconnected (light rays from one can't reach another), so there's no way we could synchronize their clocks, which means that we could add an arbitrary time offset to the clocks in each $(\theta, \phi)$ direction — which is exactly what this equation describes. In fact, for any point on $\mathscr{I}^+$ with a given $(\theta, \phi)$, the only other points on $\mathscr{I}^+$ that it is causally connected to are points with the same $(\theta, \phi)$ coordinates — but maybe different $u$ coordinates. So this explains all the supertranslations except the $\ell=0$ time offset, but that's just explained by the fact that there's no physically meaningful way to set any observer's clock; the origin of coordinates is arbitrary.
Now, this is all easiest to understand in Minkowski space using the nice, simple pictures I've shown. But it's important to remember that asymptotic flatness is a much broader idea than just Minkowski, and includes systems with much more complicated and seriously non-flat geometries inside, as well as spacetimes that may not be complete enough to draw an entire Penrose diamond like I've shown. Still, the BMS symmetries show up any time the asymptotic behavior is reasonable and flat near future null infinity. Specifically, at least some part of the limit $\mathscr{I}^+$ still exists — and near that limit, the other spacetime "looks like" the same limit of Minkowski spacetime, so the same basic rules apply.
I use the reference A. Strominger and A. Zhiboedov, "Gravitational Memory, BMS Supertranslations and Soft Theorems," for the most part here. The BMS symmetry involves the motion of test charges due to gravitational radiation from a perturbed gravitating source or black hole. The gravitational memory is due to the final positions of the masses not being the same necessarily as their initial positions. This is then a set of translations that carry information about the gravitational radiation.
To start the Bondi metric is $$ ds^2~=~-du^2~-~2dudr~+~2\gamma_{z\bar z}r^2dzd\bar z, $$ which is just the Minkowski metric for a two sphere with the $z,~\bar z$ coordinates. The term $m_B$ is the mass term, and the mass is the so called Bondi mass and source of mass-energy propagating out to ${\cal I}^+$, where the coordinate $u$ is defined. The terms $C_{zz}$ and $C_{\bar z\bar z}$ determine Weyl curvature terms for gravitational wave propagating out. An Einstein field equation is $$ D_{\bar z}^2C_{zz}~-~D_z^2C_{\bar z\bar z}~=~0, $$ which gives the simple solution $C_{zz}~=~-2D_z^2C(z,~\bar z)$. Here $C(z,~\bar z)$ is a scalar potential. The change in this potential is a change in Weyl curvature with the passage of a gravitational wave. \vskip.1in The curvature has two parts. The source terms define the Bondi mass, and the rate this changes in time, or the parameter $u$ at ${\cal I}^+$ is $$ \frac{\partial m_B}{\partial u}~=~\frac{1}{4}(D_zD_zN^{zz}~+~cc)~-~\frac{1}{4}N_{zz}N^{zz}~+~4\pi GT_{uu}(matter)|_{r~\rightarrow~\infty}, $$ where $N_{zz}~=~\partial_uC_{zz}$ is the Bondi news. The term $\frac{1}{4}N_{zz}N^{zz}$ is the flux of gravitational radiation. The stationary spacetime has $M,~Q,~J$ as Noether charges or conserved quantities. We have with that system a single set of local symmetries. These are the Lorentz or Poincare symmetries on a spatial surface, which is defined on $i_0$ since all spatial surfaces contact there. However, we know that this is an idealization of eternal black holes. The emission of gravitational radiation or the time variation of the mass, or Bondi mass, is a signature that something is more general. \vskip.1in The translations in the variables $u,~r,~z.~\bar z$, are $$ u~\rightarrow~u~+~f(z,~\bar z) $$ \begin{equation} r~\rightarrow~r~-~D^zD_zf(z,~\bar z) \end{equation} $$ z~\rightarrow~z~+~\frac{1}{r}D^zf(z,~\bar z), $$ for a function $f(z,~\bar z)$, and the translation for $\bar z$ evident. A general infinitesimal supertranslation can be easily seen to then be from these $$ \xi~=~f(z,~\bar z)\frac{\partial}{\partial u}~+~D^zD_zf(z,~\bar z)\frac{\partial}{\partial r}~-~ \frac{1}{r}\left(D^z f(z,~\bar z)\frac{\partial}{\partial z}~+~c.c.\right). $$ We may then compute the Lie derivative of the Bondi mass and Weyl curvature and the Bondi mass $$ {\cal L}_\xi m~=~f(z,~\bar z)\frac{\partial m}{\partial u,}, $$ where $\partial m/\partial u$ given above, and the Weyl curvature $$ {\cal L}_\xi C_{zz}~=~f(z,~\bar z)N_{zz}~-~2D_z^2f(z,~\bar z). $$ The supertranslations are additive, where for $\xi$ and $\xi'$ we have additive properties $\xi''~=~\xi~+~\xi'$, and these are generators of a group with $g~=~e^{p^a\xi_a}$ with the index summed over $u,~r,~z,~\bar z$. Clearly for $g'~=~e^{p^a\xi'_a}$ we have that $gg'~=~g'g$ from the additive nature of the supertranslation vectors and that metric elements are commutative. This is then an infinite dimensional abelian group.
The BMS symmetries are then a semi-direct product of the Poincare symmetries $P~=~SO(3,1)\rtimes {\cal A}$ with this group of abelian symmetries $\cal A$. The diagram below illustrates these in a conformal spacetime
Here the Poincare symmetries are at $i^0$, and the BMS symmetries manifest themselves at ${\cal I}^+$.
Now look at the metric near the event horizon, where instead of the time variable $u$ along ${\cal I}^+$ we have $v$ along the horizon with $$ ds^2~=~-\Phi dv^2 - e^{2\beta}dvdr~+~g_{AB}(dx^A~+~X^Adv)(dx^B~+~X^Bdv), $$ where $\phi$ is a potential and $(\Phi,~\beta,~g,~X)$ are arbitrary functions which means by themselves contain no particular symmetries. The coordinate $v$ is normal to $u$ along the horizon and thus metric terms with $dv$ are zero on the horizon. Near the horizon the BMS translations are $$ \xi~=~f(z,~\bar z)\frac{\partial}{\partial v}~+~\left(\int dr~-~X^A\partial_Af\right)\frac{\partial}{\partial r}~+~\left(\int dx~-~g^{AB}\partial_Af\right)\frac{\partial}{\partial X^B}, $$ where the limits on the integral are from the horizon $r_h$ to $r$ or $r~\rightarrow~\infty$ at ${\cal I}^+$. This is the near field description that shares the same information as the above translation for information sent to ${\cal I}^+$.
The correspondence between the two means there is information or data associated with the event horizon, or just above the horizon, that is encoded at ${\cal I}^+$. This is the soft hair. This is has a bearing with quotient groups. The Poincare symmetry is $P~=~BMS/\cal A$, and since $\cal A$ is an abelian group this shares some remarkable similarities to physics of the Hall effect or the fractional quantum Hall effect with quotient $[SU(2)\times SU(1,1)]/U(1)$. This soft hair is then a form of charge or BPS charge associated with the event horizon, which has physics remarkably similar to quantum Hall effect and to stick out my neck I will include symmetry protected topological states.