Given an $A_\infty$-module $M$, which is a graded module $M=\bigoplus_{k\in\mathbb{Z}}M_k$ with morphisms $m_n^M\colon A^{\otimes(n-1)}\otimes M\rightarrow M$ of degree 2-n satisfying the $A_\infty$ conditions. There is a theorem referred to by Keller in https://arxiv.org/pdf/math/9910179 that the $A_\infty$-structure of $M$ can be transferred to an $A_\infty$-structure on the homology $H(M)$ such that, as $A_\infty$-modules, $M$ is quasi-isomorphic to $H(M)$. This constructions seems to be based on homological perturbation theory and a paper cited by Keller from Stasheff. I have also seen the foundational theorem for the construction been called the homotopy transfer theorem.
Unfortunately it is very difficult for me to infer from the given resources how to explicitly construct the maps $m_n^{H(M)}$. I am not that familiar with this theory. By my limited understanding one needs to construct maps $p\colon M\rightarrow H(M)$ and $\iota\colon H(M)\rightarrow M$ such that $p\circ\iota=\text{id}$ and $\iota\circ p-\text{id}=m_1^M\circ h+h\circ m_1^M$ for some homotopy $h$ on $M$. This already confuses me. My first question is: Why should such a homotopy equivalence between the homology and $M$ even exist? And if it exists how are these maps constructed.
The even more important question: Given this data and the maps $m_n^M$ and $m_n^A$ (from the $A_\infty$-algebra itself) where can I find an explicit formula to construct $m_n^{H(M)}$. If there does not exist a nice reference maybe somebody could explain it for a layman in $A_\infty$-theory? (I am aware of some references but I could not parse what was written there.)
I would also be very happy if one could recommend a general reference for the theory of $A_\infty$-modules where things are explained in greater detail.
Thank you very much in advance.
