= Solution
The photon mean free path is
$$
\ell_\gamma=(n_e\sigma_T)^{-1}.
$$
In 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
$$
\boxed{
\lambda_d\simeq
\left(\frac{ct}{3n_e\sigma_T}\right)^{1/2}
}.
$$
In the matter era, $a\propto t^{2/3}$ and $n_e=n_{\rm rec}(a_{\rm rec}/a)^3$. Accumulating the diffusion variance while converting each displacement to comoving length gives
$$
(\lambda_d^{\rm com})^2
\simeq
\int^{t_{\rm rec}}
\frac{2c\,dt}{3n_e\sigma_Ta^2}
=\frac65
\frac{ct_{\rm rec}}
{3n_{\rm rec}\sigma_Ta_{\rm rec}^2}.
$$
Hence
$$
\boxed{
\lambda_d^{\rm com}
\simeq
\left(\frac65
\frac{ct_{\rm rec}}{3n_{\rm rec}\sigma_T}\right)^{1/2}
(1+z_{\rm rec})
}.
$$
The 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.
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