Why do Action-Angle Variables form an invariant Torus? I've been casually reading up on Hamiltonian Mechanics and integrable systems and one term that is used a lot of "invariant torus" where bounded orbits live. KAM theory is also mentioned as the reason why.
Upon searching up the wiki page for Action-Angle Variables I encounter this line:

Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables parameterize the coordinates on the torus.

However there doesn't seem to much more substantiation other that that line. So why is the "invariant torus" so insightful and important, and why do action angle variables lie on a torus?
 A: 
why do action angle variables lie on a torus?

Because, for a completely integrable Hamiltonian system, the angle variables $\phi_\sigma$ can be shown to evolve as:
$$ \phi_\sigma(t) = \nu_\sigma(\mathbf{J})t + \phi_\sigma(0). $$
Because $\phi$ is an angle, this equation describes the sweeping of an angle with angular velocity $\nu_\sigma(\mathbf{J})$ and initial condition $\phi_\sigma(0)$. $\mathbf{J}$ is just the vector of all action variables, to show that the angular velocity in general depends on them.
If there were just one angle variable (1D system), then:
$$ \phi(t) = \nu t + \phi(0) $$
describes a the motion around a circle of fixed radius.
But if there were now two angle variables, independent ("in different directions", like $p_x$ and $p_y$), then you have two circular orbits in "perpendicular" orbits:

With a little imagination , a particle that obeys evolutions in both $\phi_1$ and $\phi_2$ traces out a trajectory on a torus as shown above.
EDIT: Actually there is no imagination needed, in 2D this is exactly what happens in a tokamak nuclear fusion reactor, where the toroidal and poloidal fields play the role of $\phi_1$ and $\phi_2$, and the resulting magnetic field line is the phase-space trajectory, shown below to lie on the surface of the torus. Image from here.

This logic extends to higher dimensions, except they cannot be embedded in 3D and cannot easily be drawn anymore.
Different initial conditions would still give you torus trajectories, but with different radii (in 2D):

One says that for an integrable Hamiltonian system, phase-space (the ensemble of all possible trajectories) is foliated with non-intersecting, invariant tori.
I think the reason for the "invariant" word is that any orbit/trajectory originating on one of them, remains there indefinitely.

why is the "invariant torus" so insightful and important?

Well you have essentially proved that your phase-space is bound. That is, you can restrict your whole phase-space to sub-spaces without loss of generality, since you know your trajectories will be on these.
This is as opposed to chaos, ergodicity, etc.
Or, as an example "closer to home", the Lissajous figures for a 2D harmonic oscillator with incommensurate oscillation frequencies: these lies on still and periodic curves for a rational ratio, but fill all space for an irrational ratio.
