 Solution

The photon mean free path isIn time $t$ a photon takes about $ct/{\ell }_{\gamma }$ independent steps. Isotropy assigns one third of the mean-square displacement to any coordinate, giving the Silk damping diffusion scale

In the matter era, $a\propto t^{2/3}$ and $n_e=n_{\mathrm{rec}}(a_{\mathrm{rec}}/a{)}^3$. Accumulating the diffusion variance while converting each displacement to comoving length givesHenceThe order-one coefficient depends on the precise diffusion-length convention; the expression uses the convention stated in the question.

In a universe containing only baryons and radiation, photon diffusion erases coupled baryon-photon perturbations below this scale before recombination. Baryons begin subsequent matter-era growth from a strongly smoothed field, suppressing small-scale structure. With cold dark matter, collisionless dark-matter perturbations are not Silk damped and provide surviving gravitational wells. After recombination baryons fall into those wells, so baryonic acoustic structure is damped but small-scale total-matter structure can continue to grow.