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 $\tilde X\to X$ 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 $(\tilde X\times V)/\pi_1(X)\to X$. More explicitly, the sections of the sheaf over U can be identified with the continuous functions $f:\tilde U\to V$ satisfying $$f(\gamma x) = \rho(\gamma) f(x)$$ for $\gamma\in \pi_1(X)$. 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.
I agree that the correspondence between 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 $\tilde X\to X$ 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 $(\tilde X\times V)/\pi_1(X)\to X$. More explicitly, the sections of the sheaf over U can be identified with the continuous functions $f:\tilde U\to V$ satisfying $$f(\gamma x) = \rho(\gamma) f(x)$$ for $\gamma\in \pi_1(X)$. This sheaf can be checked to be locally constant.