What is physically happening in a medium in which an evanescent wave is propagating? Let's consider for instance a metal, for which the dielectric function reads:
$\epsilon = 1 - \frac{\omega_{p}^{2}}{\omega^{2}}$
where $\omega_{p}$ is the plasma frequency. The dispersion relation for an electromagnetic wave is then:
$\omega^{2} = \omega_{p}^{2} + k^{2} c^{2}$
For $\omega < \omega_{p}$, the wavenumber is imaginary and such a wave cannot propagate inside the medium, instead getting damped exponentially as an evanescent wave.
But what is physically happening inside the material that prevents the wave from propagating? How are the electrons of the medium responding to such a wave and why does that behavior stop the propagation of the wave and reflect it back instead?
 A: You are correct that the wavenumber becomes purely imaginary.  As you approach $\omega = \omega_{pe}$ from the $\omega > \omega_{pe}$ side, the phase speed diverges (i.e., $\tfrac{ \omega }{ k } \rightarrow \infty$) and the group speed goes to zero.  Inside the region where $\omega < \omega_{pe}$, the fields do not oscillate like they do outside (if the wave is evanescent).  This is because the conduction current balances the displacement current when $\omega = \omega_{pe}$.  That is, the ratio of the conduction to displacement current is given by:
$$
\left( \sum_{s} \frac{ n_{s} \ e^{2} }{ -i \ m_{s} \ \omega } \right) \left( \frac{ 1 }{ -i \ \varepsilon_{o} \ \omega } \right) = - \frac{ \omega_{pe}^{2} }{ \omega^{2} } \tag{0}
$$
where $n_{s}$ is the number density [e.g., # cm-3] of species $s$, $m_{s}$ is the rest mass [e.g., kg] of species $s$, $e$ is the fundamental charge, and $\varepsilon_{o}$ is the permittivity of free space.
As you can see, when $\omega \gg \omega_{pe}$, the conduction current is tiny compared to the displacement current.  However, as $\omega \rightarrow \omega_{pe}$, the conduction current becomes more and more important (i.e., the electromagnetic fluctuations couple to the plasma).  Eventually, the fields can no longer cause charged particles to oscillate, so they exponentially decay with increasing distance into the medium.
The more useful aspect of this interaction is the reflection/blocking part.  That is, we can use our knowledge of this to determine origins of radio signals based on the receiver location.  We do this with solar radio bursts and ionospheric interference, reflection, and conduction experiments.  Some radio communications intentionally reflect radio signals off the ionosphere, called skywave or skip.
