I have a theoretical question about comparing two objects that I have recently come across.

For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over $k$. One can construct the stack $BG$ as a fibered category in groupoids and as a simplicial scheme $(BG)_n=G^n$ (with certain face and degeneracy maps which I don't specify). What is the precise relation between these two versions of $BG$? Can I obtain one from the other?

I think I might have heard that "the two constructions are equivalent because the simplicial $BG$ has no higher homotopy groups". Does this make sense? How does this implication work?

Any answer that could help me better understand the relation between these two $BG$s is very welcome.