Let $M$ be a nice connected topological space (I'm actually interested in manifolds) with base point $p$ and let $\pi: E \to M$ be a fibration. Then chains on the fiber $F$ at $p$, $C_*(F)$, become a dg-module over $C_*(\Omega M)$, the based loop space at $p$. Then if I'm interpreting Blumberg–Cohen–Teleman correctly (Open-closed field theories, string topology, and Hochschild homology line just after Lemma 2.5), there is an isomorphism $$ C_*(E,k) \cong k\otimes^{L}_{C_*(\Omega M)} C_*(F) $$

I'm wondering if anyone can tell me a reference which explains this kind of formula in more detail? Perhaps, asked differently, let's say that I'm comfortable with Hatcher's book. I also can follow Moore's proof in Algèbre homologique et homologie des espaces classifiants which should correspond to the special case $M=BG$ using the fact that $\Omega BG \simeq G$.