sheaf theory - group cohomology and cohomology of classifying space

Let $G$ be a discrete group, and $BG$ is the classifying space.
It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $tilde{M}$, which is the associated sheaf of $M$.

Can someone explain how these two cohomologies are related?

• Next gn.general topology - Which is the correct ring of functions for a topological space?
• Previous lo.logic - What is a logic ?