The geometrical interpretation of the Poisson bracket

The importance comes from the equations of motions: $$ \dot{q}=\frac{\partial H}{\partial p}\, ,\qquad \dot{p}=-\frac{\partial H}{\partial q} $$ which can be rewritten as $$ \dot{q}=\{q,H\}\, ,\qquad \dot{p}=\{p,H\} $$ In particular, for an arbitrary function of $f(p,q)$, we have $$ \frac{d}{dt}f(p,q,t)=\{f,H\}+\frac{\partial f}{\partial t}\, . $$ Geometrically, changes of coordinates in phase space (i.e. canonical transformation) preserve the Poisson bracket, i.e. the transformation $(q,p)\to (Q(q,p),P(q,p)$ is such that $$ \{q,p\}=\{Q,P\}=0\, , $$ and so in this sense, and thinking of $q$ and $p$ as "basis vectors", canonical transformation preserve the Poisson bracket much like rotation preserve the dot product between two vectors. In this interpretation a quantity is conserved (i.e. $\dot f(q,p,t)=0$) when it is Poisson-orthogonal to the Hamiltonian, for instance.

The Poisson bracket is the bracket of a Lie algebra defined by the symplectic 2-form.

That's a lot to unpack so let's go through it slowly. A 2-form $\omega$ is an anti-symmetric two-tensor $\omega_{\mu\nu}$. If $\omega_{\mu\nu}x^\nu \neq 0$ at points where $x^\nu \neq 0$, then $\omega$ is said to be non-degenerate. So $\omega$ is like the metric tensor, except it's anti-symmetric instead of symmetric. To be symplectic, the "curl" ("exterior derivative") of $\omega$ should also be zero, $(d\omega)_{\mu\nu\rho} = \partial_{[\mu} \omega_{\nu\rho]} = 0$ where the brackets indicate complete anti-symmetrization.

Since $\omega_{\mu\nu}$ is non-degenerate, like the usual metric tensor, it defines an isomorphism between vectors (index up) and one-forms (index down). If $f$ is a scalar function, then $\partial_\mu f$ is naturally a one-form. With this isomorphism, we can define an associated vector field $X_f^\mu$. Note that in terms of concrete components, what this is does is quite different from the usual operation of raising and lowering indices in relativity. Viz., in 2 dimensions, any symplectic 2-form can be represented by the matrix $\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}$ so that if $\partial_\mu f$ has components $(a, b)$, $X_f^\mu$ has components $(b,-a)$.

Because we can get one-forms from scalars by taking the gradient, we can define an operation on scalars $f,g$ as $(f,g) \mapsto \omega(X_f, X_g)$. One can verify that this operation is linear in both arguments, anti-symmetric, and satisfies the Jacobi idenitity (because of the requirement that the exterior derivative $d\omega$ vanishes), so it defines a Lie algebra.

If you use the matrix representation of $\omega$ above and that in coordinates $p,q$, the components of $\partial_\mu f$ are $(\partial f/\partial p, \partial f/\partial q)$, then you can work out that this coincides with the usual definition of the Poisson bracket in coordinates. The extension to $2n$ dimensions with coordinates $p_i, q_i,\, i = 1,\ldots,n$ is found by replacing $1$ by the $n\times n$ idenity matrix in the matrix above. A theorem by Darboux says that this can always be done locally.

The canonical reference for this is V. I. Arnold, Mathematical Methods of Classical Mechanics.

Combine $q,\,p$ into a single vector $y$, said to live in phase space (which of course is even-dimensional). The PB is $\partial_i f\omega^{ij}\partial_j g$ where $y_i$ is either a component of $q$ or $p$ and $\omega^{ij}\partial_j g$ is the symplectic gradient, and the matrix/$2$-form $\omega$ is left as an index to the reader. The geometric interpretation is that phase space is a symplectic manifold, a manifold equipped with a certain kind of $2$-form.