I just read about the definitions about torsor of sheaf of groups and get a bit confused. 
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a similar weak equivalence $\Omega B G\simeq G$ for $G$ a sheaf of group? 


Why is there an equivalence $\Omega BTors(G)\simeq G$? 



$G$-torsor for topological space compared to that for sheaf of groups
The notion of $G$-torsor for a topological space is the same as the sheaf-theoretic definition, once you are familiar with the language of site.


Your desired weak equivalences exist, in which the looping need (right) derived (w.r.t. the local model structure), they follow from Jardine's *Local Homotopy Theory*, Proposition 9.26 and Corollary 9.27. I suggest you work out these as a good exercise.


