1. Is there a low level "homotopical" description of cohomology with local coefficients? Similar to the identification of ordinary singular cohomology with the homotopy classes of maps to the Eilenberg-MacLane space $K(G,n)$.

I saw the page in nLab, but it's written in the language I don't understand. I also found this more general question with some nice clarifications by Urs.

I'm guessing that it means the following. Given an action of $\pi_1(X)$ on an (abelian) group $G$, we get a bundle over $X$ with fiber $G$, from which we get a bundle over $X$ with fiber $K(G,n)$ (or whatever its nonabelian analog). Then $H^n(X;G)$ is the homotopy classes of sections of the latter bundle. Is that interpretation correct?

2. Is there a geometric object we can associate to the cohomology in the case of $n=1$? Similarly how we can use an element of untwisted cohomology to make a $G$-bundle, is it possible by taking an element of twisted cohomology to glue some kind of space (perhaps not from trivial pieces)?

Of course, I would be interested if the second question has an answer for $n>1$, but I don't even know what it is for the untwisted theory.