reference request - Why are local systems and representations of the fundamental group equivalent

I agree that the correspondence between representations of the fundamental group(oid) and locally constant sheaves is not very well documented in the basic literature. Whenever it comes up with my
students, I end up having to sketch it out on the blackboard. However, my recollection is that Spanier's Algebraic Topology gives the correspondence as a set of exercises with hints. In any case, one direction is easy to describe as follows. Suppose that $X$ is a good connected space X (e.g. a manifold). Let $tilde Xto X$ denote its universal cover. Given a representation of its fundamental $rho:pi_1(X)to GL(V)$, one can form the sheaf of sections of the bundle $(tilde Xtimes V)/pi_1(X)to X$. More explicitly, the sections
of the sheaf over U can be identified with the continuous functions $f:tilde Uto V$ satisfying
$$f(gamma x) = rho(gamma) f(x)$$
for $gammain pi_1(X)$. This sheaf can be checked to be locally constant.
Essentially the same procedure produces a flat vector bundle, i.e. a vector bundle with locally constant transition functions. This is yet another object equivalent to a representation of the fundamental group.

With regard to your other comments, perhaps I should emphasize that the Narasimhan-Seshadri correspondence is between stable vector bundles of degree 0 and irreducible
unitary representations of the fundamental group. The bundle is constructed as indicated above.
Anyway, this sounds like a good Diplom thesis problem. Have fun.