$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$.  A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

You have to careful here, because it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles.