Why does the Stinespring dilation of stabilizer operations have the form $\mathcal{E}(\rho) = tr_E(U \rho \otimes \rho_E U^\dagger)$?

Question

Why does the Stinespring dilation of a stabilizer operation have the form

$\mathcal{E}(\rho) = tr_E(U \rho \otimes \rho_E U^\dagger)$

where $U$ is a Clifford unitary and $\rho_E$ is a stabilizer state?

Also, why does tracing out only take stabilizer states to other stabilizer states?

I am looking for an answer that applies to stabilizer subtheories for systems of any dimension.