Landau damping seems mysterious because it is derived from completely reversible equations, yet it gives a "damping" behaviour. But the funny thing is that when you go through the equations you find even an "antidamping" exponentially growing solution$^1$!
The reason for this is that the Landau damping isn't really a damping caused by particle collisions. It is caused by the electrons in the plasma "getting out of sync" and causing the wave to disappear because every electron creates a slightly different wave at a different phase. This can be shown through an ensemble of harmonic oscillators representing the electrons.
The exponentially growing solution corresponds to the oscillating electrons on the contrary "getting into sync" and superposing again into a macroscopic wave. However, the original Landau analysis is done in linear order, so the exponential growth of the wave will be altered as soon as the wave grows large enough.
Landau damping is an "entropy preserving process" in the sense that the single electrons do not get randomized but the direction of the wave is still reflected in their movement - there is a slight hidden anisotropy in the plasma in the direction of the damped wave even after it's disappearance. This anisotropy is indeed macroscopic in the sense of a deviation from the isotropy of the average velocities of particles.
The question is whether to count these hidden oscillations into heat gained by the plasma or not. If not, then entropy was indeed preserved, but if yes, entropy grows just by the decoherence of oscillations. Truth is entropy tends to be badly defined for non-equilibrium ensembles, so this is why Cedric Villani uses the quotation marks in talking about an "entropy-preserving process".
However, in practice real damping and collisions are happening along Landau damping and this takes quickly care of the hidden anisotropy.
$^1$ This is done by taking the lower counterpart of the "Landau contour", see the original paper by Landau On the vibration of the electronic plasma..