How does the emf "know" what the magnetic flux is?
Suppose you have an infinitely long solenoid made of "stacked" loops of radius $a$. The magnetic field is then truely 0 outside, and all the field lines are confined inside the solenoid. Then how can an exterior loop (of radius $b \ge a$) encircling the solenoid could "know" that there's actually a changing magnetic field inside the solenoid, since it is completely confined ?
First, the magnetic flux is defined by this expression : \begin{equation}\tag{1} \Phi_B \equiv \int_{\mathcal{S}_{\text{sol}}} \vec{B} \cdot d\vec{S}, \end{equation} Where $\mathcal{S}_{\text{sol}}$ is the solenoid transverse aera. Since the field vanishes outside the solenoid, you can use the exterior loop aera $S_{\text{loop}}$ instead, and express the magnetic field in terms of the magnetic vector-potential : $\vec{B} = \vec{\nabla} \times \vec{A}$ : \begin{equation}\tag{2} \Phi_B = \int_{\mathcal{S}_{\text{loop}}} \vec{B} \cdot d\vec{S} = \int_{\mathcal{S}_{\text{loop}}} (\vec{\nabla} \times \vec{A}) \cdot d\vec{S}. \end{equation} By Stokes theorem, you then have the magnetic flux expressed as a line integral around the loop : \begin{equation}\tag{3} \Phi_B \equiv \oint_{\mathcal{C}_{\text{loop}}} \vec{A}_{\text{outside}} \cdot d\vec{\ell}. \end{equation} The vector-potential doesn't vanish outside the solenoid (it must be continuous across the solenoid's boundary) : \begin{align}\tag{4} \vec{A}_{\text{inside}} &= \frac{1}{2} \; \vec{B} \times \vec{r}, &\vec{A}_{\text{outside}} &= \frac{a^2}{2 \, \rho^2} \; \vec{B} \times \vec{r}, \end{align} where $a$ is the radius of the solenoid and $\rho$ is the cylindrical variable. $\vec{r}$ is the vector position of any point in space, and $b \ge a$ is the loop radius. Using this vector-potential, it is very easy to verify that \begin{align} \vec{\nabla} \times \vec{A}_{\text{inside}} &= \vec{B}, &\vec{\nabla} \times \vec{A}_{\text{outside}} &= 0, \end{align} and expression (3) gives $\Phi_B = \pi B \, a^2$.
So, the loop doesn't feel the magnetic field itself, but it can interact with the vector-potential. The e.m.f is the time derivative of the flux : \begin{equation}\tag{5} \mathscr{E} = -\: \frac{d \Phi_B}{d t} = -\: \frac{d}{d t}(\pi B \, a^2) = -\: \pi \dot{B} \, a^2. \end{equation} Now, the e.m.f itself is defined as the line integral of the electric field induced on the loop by the time-varying magnetic field inside the solenoid : \begin{equation}\tag{6} \mathscr{E} \equiv \oint_{\mathcal{C}_{\text{loop}}} \vec{E} \cdot d\vec{\ell} = \pm \, E \; 2 \pi b, \end{equation} Then we get $E(t) = \frac{a^2}{2 \, b} \; |\, \dot{B} \,|$ on the loop, or \begin{align}\tag{7} \vec{E}_{\text{inside}}(t, \, \vec{r}) &= -\: \frac{\partial }{\partial t} \, \vec{A}_{\text{inside}} = -\: \frac{1}{2} \; \frac{\partial \vec{B}}{\partial t} \times \vec{r}, \\[18pt] \vec{E}_{\text{outside}}(t, \, \vec{r}) &= -\: \frac{\partial }{\partial t} \, \vec{A}_{\text{outside}} = -\: \frac{a^2}{2 \, \rho^2} \; \frac{\partial \vec{B}}{\partial t} \times \vec{r}, \tag{8} \end{align} which agrees with Maxwell's equation : \begin{align}\tag{9} \vec{\nabla} \times \vec{E}_{\text{inside}} &= -\: \frac{\partial \vec{B}}{\partial t}, \\[18pt] \vec{\nabla} \times \vec{E}_{\text{outside}} &= 0. \tag{10} \end{align} Take note that $\vec{\nabla} \cdot \vec{E} = 0$ everywhere (do all the detailled calculations to verify this !). So the conclusion is that the loop do indirectly feel the magnetic field of the solenoid with the help of its vector-potential, outside the solenoid.
Complement : Take note that $\vec{B}$ must be varying very slowly, or vary linearly with $t$, or else there will be some electromagnetic waves outside the solenoid ! We have \begin{equation}\tag{11} \vec{\nabla} \times \vec{B}_{\text{outside}} = 0 = \mu_0 \, \vec{J}_{\text{outside}} + \frac{1}{c^2} \, \frac{\partial}{\partial t} \, \vec{E}_{\text{outside}}. \end{equation} The current density $\vec{J}$ vanishes inside and outside the solenoid (and it is singular on its boundary !). Then equation (11) and expression (8) give \begin{equation}\tag{12} 0 = \frac{1}{c^2} \, \frac{\partial}{\partial t} \, \vec{E}_{\text{outside}} \propto \frac{1}{c^2} \, \frac{\partial^2 \, \vec{B}}{\partial t^2}. \end{equation} This is also true inside the solenoid.
Based on the Faraday’s induction law: $$ \mathcal{E}_{EMF} = \oint_{C_1} E \cdot dl = -\frac{d \Phi}{d t} ~, $$ where $E$ is the induced electric field and $$ \frac{d \Phi}{d t} = \frac{d B}{d t}\pi r_2^2. $$ Then left-hand-side of this equation, assuming that the integral is over a circular curve $C_1$ with a radius $r = r_1$ (because you are calculating the field at that distance), and that $E$ is uniform over that curve, we have: $$ E \times \text{circumference of a circle of radius } r_1 = E \times 2\pi r_1 ~. $$ so that $$ \begin{align} E\times 2\pi r_1 &= -\frac{dB}{dt}\pi r_2^2 \\ \implies E &= -\frac{dB}{dt} \left(\frac{r_2^2}{2r_1}\right) \end{align} $$ Therefore, with the right calculations you see that an electric field is induced by the change of $B$ in the solenoid, this electric field depends on the shape of both the ring and the solenoid ($r_1$ and $r_2$) and causes the electrons to move on the larger ringe; hence, an electric current is induced in that ring.
The analytical solution to the problem of two loops (a solenoid inside a conducting ring) is given by Armin R (here) which shows how the induced electric field in the larger ring depends on the parametres of the problem (the radii).
Here, I attempt to extend Armin R’s contribution by answering the essential (more interesting) part of the question,
How does the large loop even “know” what the flux is, while we assume that the magnetic field $B$ due to the solenoid is negligible outside of the smaller cross section of the solenoid, so that there is no field actually going through the material of the larger conducting loop. ... It seems there is no way for the loop to “feel” the magnetic flux.
To understand how the larger ring “feels” the changing magnetic flux, one should remind that the Maxwell equations which describe the dynamics of electromagnetic fields, predict that any change in the electromagnetic field at some point would propagate with the velocity of light throughout the space. This happens, e.g., in an antenna where the oscillation of electric charge produces waves which we can “feel” (detect) from far away.
The same thing happens in this case, but such ‘details’ or complexities are eliminated by the simplified scenario of the problem; namely, the problem is given in the context of steady-state electrodynamics where the transient dynamics has disappeared and one does not consider how the effect of the change in the localized magnetic field $B$ propagates in space-time to reach the larger ring. This becomes clear by noticing that in the analytical solution by Armin R, everything is instantaneous,
$$
E(\color{red}{t}) \propto -\frac{d B(\color{red}{t})}{d t} ~;
$$
that is, changing $B$ changes $E$ instantaneously (with no time delay) and this apparently $^\ast$ contradicts the relativistic nature (Lorentz invariance) of Maxwell's equations – the finite time needed for the changes to propagate.
So, the actual (full) scenario is that when one changes the localized field $B$, this effect propagates (like waves) in space-time with the velocity of light and ultimately, after $\Delta t \sim \frac{|r_2 - r_1|}{c}$, reaches the charge carriers (electrons) in the larger ring. In this way, the ring “feels” the effect of the change in the magnetic field of the solenoid (despite the magnetic field being localized).
I think considering the Liénard–Wiechert potentials that describe the classical electromagnetic effect of a moving electric point charge, in terms of relativistic, time-varying electromagnetic fields will be enlightening regarding the current question.
$^\ast$ This is only an apparent (not actual) contradiction, as noted in a comment by John Duffield: “one should properly speak of the electromagnetic field $F_{\mu \nu}$ rather than $E$ or $B$ separately” [$\S$ 11.10 of Jackson’s “Classical Electrodynamics”]. Here, I intend to emphasize the instantaneity.