Here is the way I think of the correspondence between locally constant sheaves and representations of the fundamental group, and how I like to tell my students about it when I introduce it in class (the expected background being the fundamental group, the theory of covering spaces, and the notion of sheaf).
Sheaves as sheaves of sections of étalé spaces
First, there is a correspondence
$$
\mathrm{Sh}_X:=\{\mathrm{sheaves\ of\ sets\ on}\ X\}
\leftrightarrow
\mathrm{Et}_X := \{\mathrm{local\ homeomorphisms}\ p:Y\longrightarrow X\}
$$
sending a sheaf $\mathcal{F}$ on $X$ to the étalé space
$$\mathrm{Et}(\mathcal{F}) := \bigsqcup_{x\in X} \mathcal{F}(x)\, ,$$ where $\mathcal{F}(x) := \varinjlim_{U\ni x} \mathcal{F}(U)$ is the stalk of $\mathcal{F}$ at $x$, and an étalé space $p:Y\longrightarrow X$ to the sheaf of (continuous) sections $$U\longmapsto\Gamma_Y(U):=\{s:U\longrightarrow Y\ |\ p\circ s=\mathrm{id}_U\}.$$
This is useful already in the basic theory of sheaves, for instance to construct the sheaf associated to a pre-sheaf. One possible reference is:
Sections of covering spaces
Next, under this correspondence, locally constant sheaves correspond precisely to covering spaces:
$$
\mathrm{Loc}_X:=\{\mathrm{locally\ constant\ sheaves\ of\ sets\ on}\ X\}
\leftrightarrow
\mathrm{Cov}_X := \{\mathrm{covering\ map}\ p:Y\longrightarrow X\}
$$
Indeed, a locally constant sheaf is locally isomorphic to the sheaf of continuous sections of a product space $X\times F$, where $F$ is a discrete topological space (in particular, such sections are locally constant maps, with values in $F$).
The upshot of working with a covering map $p:Y\longrightarrow X$ is that (since $X$ is nice) we can lift paths in $X$ (and homotopies between them) to $Y$, in a unique manner. In particular, there is a well-defined map
$$\rho: \pi_1(X,x) \longrightarrow \mathrm{Aut}(Y_x)$$ (where $Y_x:=p^{-1}(\{x\})$ is the fibre of $p$ above $x$) defined by sending the homotopy class of a loop $\gamma:[0;1]\longrightarrow X$ at the base point $x\in X$ to the bijective transformation $$\rho(\gamma): (y \in F) \longmapsto \widetilde{\gamma}^{(y)}(1) \in F$$ where $\widetilde{\gamma}^{(y)}:[0;1]\longrightarrow Y$ is the unique continuous map such that $p\circ\widetilde{\gamma}^{(y)}=\gamma$ and $\widetilde{\gamma}^{(y)}(0)=y$.
The correspondence
With an appropriate convention on the composition of paths in $X$, the map $\rho$ becomes a group morphism. So, given a discrete topological space $F$, the choice of base point $x\in X$ induces a map
$$\Phi:\{\mathrm{covering\ spaces\ of}\ X\ \mathrm{with\ fibre}\ F\} \longrightarrow \mathrm{Hom}\big(\pi_1(X,x);\mathrm{Aut}(F)\big)$$
The converse map
$$\Psi: \mathrm{Hom}\big(\pi_1(X,x);\mathrm{Aut}(F)\big) \longrightarrow \{\mathrm{covering\ spaces\ of}\ X\ \mathrm{with\ fibre}\ F\}$$
is the map defined in Donu Arapura's answer: a group morphism $\rho: \pi_1(X,x) \longrightarrow \mathrm{Aut}(F)$ is sent to the covering space
$$(\widetilde{X}\times F)\,/\,\pi_1(X,x) \longrightarrow \widetilde{X}\,/\,\pi_1(X,x) = X$$
where, for $X$ nice, $\widetilde{X}$ is the universal covering space of $X$ (determined up to canonical isomorphism by the choice of the base point $x\in X$) and $\gamma\in\pi_1(X,x) \simeq \mathrm{Aut}_X(\widetilde{X})$ acts on $(\xi,v)\in(\widetilde{X}\times F)$ via $$\gamma\cdot(\xi,v) := \big(\gamma\cdot\xi, \rho(\gamma)\cdot v\big).$$
Local systems of vector spaces
Once it is checked that this is indeed a covering space of $X$ with fibre $F$, it remains to prove that the maps $\Phi$ and $\Psi$ are indeed inverse to each other.
This is compatible with the notion of isomorphisms of covers and equivalence of representations, so it provides a bijection
$$\check{H}^1\big(X;\mathrm{Aut}(F)\big) \simeq \mathrm{Hom}\big(\pi_1(X,x);\mathrm{Aut}(F)\big)\, \big/\, \mathrm{Aut}(F)$$
where $\check{H}^1\big(X;\mathrm{Aut}(F)\big)$ is the set of isomorphism classes of locally trivial bundles with (discrete) fibre $F$ and structure group $\mathrm{Aut}(F)$ over $X$.
When $F=V$ is a finite-dimensional vector space equipped with the discrete topology, you can restrict the above to the subgroup $\mathbf{GL}(V) \subset \mathrm{Aut}(V)$ and obtain a correspondence
$$\{\mathrm{local\ systems\ of}\ \mathit{vector\ spaces}\ \mathrm{on}\ X\}
\leftrightarrow
\{\mathit{linear}\ \mathrm{representations\ of}\ \pi_1(x,x)\}.$$
The notation $\check{H}^1\big(X;\mathbf{GL}(V)\big)$ is commonly to designate the set of isomorphism classes of flat vector bundles with fibre $V$ on $X$ (flat in the sense that the transition functions $$g_{U_2U_1}:U_1 \cap U_2 \longmapsto \mathbf{GL}(V)$$ are locally constant on the open set $U_1 \cap U_2\subset X$).
To go further, if the topological space $X$ is a real or complex manifold $(X,O_X)$, the correspondence between locally constant sheaf of vector spaces and linear representations of the fundamental group can also be phrased in terms of vector bundles equipped with an integrable connection.
Namely, as in this answer to a related question on MO, a local system of vector spaces $\mathcal{V}$ is sent to the vector bundle ($=$ locally free $O_X$-module) with integrable connection
$$(\mathcal{E}:=
O_X \otimes_{\mathrm{Cst}_X} \mathcal{V}\, ,\ \nabla:= d \otimes\mathrm{id})$$
where $\mathrm{Cst}_X\subset O_X$ is the sheaf of locally constant functions on $X$ (with values in $\mathbb{R}$ or $\mathbb{C}$) and $d:O_X\longrightarrow \Omega^1_x$ sends a function $f\in O_X(U)$ to the $1$-form $df\in\Omega^1_X(U)$, so
$$\nabla:\mathcal{E}\longrightarrow \Omega^1_X \otimes_{O_X} \mathcal{E}$$
is indeed a linear connection on $\mathcal{E}$ (=it satisfies the Leibniz identity $\nabla(f\cdot s) = df\otimes s + f\cdot \nabla s$ for all open set $U\subset X$, all section $s\in\mathcal{E}(U)$ and all function $f\in O_X(U)$).
In this picture, the local system of vector spaces $\mathcal{V}$ is viewed as a locally free $\mathrm{Cst}_X$-module, which is why the operation $$\mathcal{V}\longmapsto O_X \otimes_{\mathrm{Cst}_X} \mathcal{V}$$ makes sense.
The converse map sends the vector bundle with integrable connection $(\mathcal{E},\nabla)$ to the sub sheaf $$U\longmapsto\mathcal{E}^\nabla(U) := \{s\in \mathcal{E}(U)\ |\ \nabla s=0\},$$ which is a locally constant sheaf (the sheaf of locally constant sections of $\mathcal{E}$, which is locally isomorphic to the sheaf of differentiable sections of $X\times V$, where $V$ is a finite-dimensional real or complex vector space, endowed here with its usual topology).
