The important point of the proof is that either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.
EDIT: Brian Conrad points out below that while this is a fairly complete sketch in the smooth case, it requires more work in the singular case.
If your question concerns - as mentioned in one of your comments - if there is any relationship between them, then a very beautiful connection exists in what is called geometric class field theory:
namely classical number theoretic class field theory concentrates around what is called Artin Reciprocity, which establishes an isomorphism for a number field $K$ and its ring of integers $\mathcal{O}_{K}$ an isomorphism $Pic(Spec(\mathcal{O}_{K})) \cong \pi_{1}^{ab}(Spec(\mathcal{O}_{K}))$ between the Picard group and the abelianized etale fundamental group (it is a geometric reformulation of classical Artin reciprocity). We can see it as a special case of one-dimensional class field theory and the question arises naturally if we can extend somehow this correspondence for higher dimensions (and also for other one dimensional schemes). There are different approaches (K-theory, cycle theory, geometric Langlands) but the main cornerstones are the following:
Bloch-Kaito-Saito Theorem: Let $X$ be a regular, connected, projective scheme over $Spec(\mathbb{Z})$, then there exists also a reciprocity map $Pic(X) \rightarrow \pi_{1}^{ab}(X)$ which is an isomorphism if in addition $X$ is flat over $Spec(\mathbb{Z})$. If $X$ factors through a finite field $k=\mathbb{F}_{q}$ then the reciprocity map is injective and with cokernel $\widehat{\mathbb{Z}}/\mathbb{Z}$.
Also for curves over finite fields there exists a correspondence, namely if $C$ is a smooth, projective, geometrically irreducible curve over a finite field $k$, then there exists a reciprocity homomorphism $Pic_{C}(k) \rightarrow \pi_{1}^{ab}(C)$ which induces an isomorphism on the degree zero parts $Pic_{C}^{0}(k) \rightarrow \pi_{1}^{ab}(C)^{0}$, where the degree maps are the obvious maps to $\mathbb{Z}$ and $\widehat{\mathbb{Z}}$ resp.
Also if $S \subset C$ is a finite set of points of a smooth, projective, geom. irreducible curve $C$ over a finite field, then there is a ramified version of the previous reciprocity, namely between $Pic_{C,S}$ (which is isomorphism classes of line bundles together with fixed isomorphisms of the stalks at every point in $S$) and the abelianization of the tame fundamental group of $U:=C \setminus S$.
Some reference:
http://epub.uni-regensburg.de/13979/1/MP92.pdf
and then it gives many other references and so on...
Best Answer
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.
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.