Configuration spaces to moduli spaces

Question

In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly.

Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \partial$). (In the paper, $F_{g,2}$ is a surface of genus $g$ with two boundary circles)

This seems to be a well known result, could you let me know some references please?

Additionally, I'm confused with the nerve of a category with the classifying space. (Correct me if I'm wrong, but I've heard somewhere that they were homotopically equivalent. Is it correct?)

Thanks in advance.