What exactly is Landau Damping?

This wikipedia article explains the physical picture rather well. Landau damping results from the energy exchange between the electromagnetic wave and the plasma particles, its main feature being the dependence on the distribution of the particles.

As an analogy, perhaps a bit far-fetched, one could compare Landau damping with the interaction of an electromagnetic wave with an active environment of a laser. As long as the population of the ground state is higher than that of the excited state, we always have more photons absorbed than emitted. However, if the populations of the ground and the excited states are inverted, the opposite happens and we achieve the amplification of the signal.

Such thinking about damping and amplification is rather comonplace - Landau's main achievement is describing (and predicting) it in the context of plasma. Note that this was quite commongly and literally used in various kinds of bulb devices throughout the XXth century, but less familiar nowadays (which is why the laser analogy might be more understandable.)

A way of intuitive thinking about Landau daming is from the point of view of the electrostatic wave (or Langmuir oscillation). This will also clarify a bit why the whole mathematical apparatus is needed.

Short Intro to Electrostatic Waves

Without a magnetic field, in an electrostatic field you can consider the Vlasov equation and derive a dispersion relation for a wave, $$ 1 + frac{omega_e^2}{k^2} int frac{partial v_x g(v_x)}{omega/k-v_x} dv_x = 0, $$ where $omega_e$ is the electron plasma frequency, $v_x$ is the x-velocity coordinate and $g(v_x)$ is a 1d distribution that relates to the background distributions of all species (where $v_y, v_z$ is integrated out).

You can see that this is quite complicated and contains a pole at $k v_x = omega$. Anyway, using a bit of hand-wavy assumptions (the pole is away from a region where the distributions are non-zero) you can integrate this and (after some manipulation) arive at the Langmuir wave dispersion relation (only taking electrons into account into $g$), $$ omega^2 = omega_e^2 + 3 k^2 v_e^2, $$ where $v_e$ is the mean velocity of the electron species (1st moment). This is basically a simple oscillation of the electrons with a frequency slightly higher than the electron plasma frequency.

Landau Damping

So now so good, the part where Landau damping comes into play is where you ask yourself if the integral should need a proper treatment. Then you have to make the integration in the complex plane and so on. You will get a modification to the above dispersion relation, but the main point is, that it becomes complex with the imaginary part or damping, $$ gamma propto partial v_x g(v_x) Bigg|_{v_x = omega / k}. $$ This means, that there is damping which depends on the slope if the distribution $g(v_x)$. Here come the metaphors with e.g. the surfers into play. A surfer slower than a wave is sped up until he reaches the bottom of the wave, if he is faster the other way round. The surfer shoul be associated with a particle, it is loosing or gaining energy from the wave, depending on its velocity compared to the wave.

One important remark here is, that we used the Vlasov equation only, i.e. no collisions. Thus, Landau damping is classified as collisionless damping.

References

I mostly recommend Nicholson, Introduction to Plasma Theory (although it is quite old and possibly impossible to get nowadays). You may also find more in Chen, Introduction to Plasma Physics and Controlled Fusion (including the surfer).