Why string amplitude could be computed by path integral of string field theory?

I am trying to understand BCOV’s paper: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes.

In this paper, it was shown that the higher genus string amplitude for B-model could be computed by the path integral of its string field theory, which is Kodaira-Spencer theory introduced in the paper above.

My question is, are there any nice explanations, showing why string amplitude could be computed by path integral of string field theory? What’re the exact correspondences between higher loop path integral for string field and higher genus terms in string theory?

I am a math student and familiar with Feynman path integrals, some basic facts about CFT, BRST quantizations e.t.c. For path integral, I know that if $Z_S$ is the portion function w.r.t action $S$, then we can find a new “effective” action $S_{eff}$ (which is a “deformation” of $S$) such that
$(ln(Z_{S_{eff}} /Z_0))_0 = ln(Z_S/Z_0)$. Is it related to my question?