In Spanier AT (p.325), there is a notion $\Gamma(\mathcal{U})$ called the module of compatible $\mathcal{U}$ families of $\Gamma$ where $\mathcal{U} = \{U\}$ is a collection of open sets and $\Gamma$ is a presheaf of modules on $X$.
Definition. A compatible $\mathcal{U}$ family of $\Gamma$ is an indexed family $\{\gamma_U\in\Gamma(U)\}_{U\in\mathcal{U}}$ such that $\gamma_U\mid U\cap U' = \gamma_{U'}\mid U\cap U'$ for $U,U'\in\mathcal{U}$.
What exactly $\Gamma(\mathcal{U})$ is? I don't think it's a graded module but something else. Because of the lack of understanding, I can't even understand simple example:
The constant presheaf $G$ defined by a module $G$ is not generally a sheaf [if $U$ is a disconnected open set, $G(U)\not\simeq\hat{G}(U)$].
Could you help?
Given an $R$-module $G$, the constant presheaf $G$ on $X$ assigns to every nonempty open $U\subset X$, the module $G$.
A compatible $\mathcal{U}$ family of $\Gamma$ is a pack of local sections that glue together. The set of those families $\Gamma(\mathcal{U})$ inherits a module structure from $\Gamma$ (just add and multiply the sections at each open set). Moreover, applying restrictions on each open set makes it a direct system of modules.
Now the direct limit of this system is obtained as the quotient module of: $$\bigsqcup_{\mathcal{U}}\Gamma(\mathcal{U}) $$ where two families $\{\gamma_U\}\in\Gamma(\mathcal{U})$ and $\{\delta_U\}\in\Gamma(\mathcal{V})$ are equivalent if there is a common refinement of $\mathcal{U}$ and $\mathcal{V}$ on which they restrict to the same sections.
The example you mentioned is important to see what's happening. You start with the presheaf of constant functions with values in $G$, which is not a sheaf because constant functions won't glue to another constant function, if your base space has more than one connected component. This is what taking the direct limit above achieves: it adds those sections that should have glued together the constant functions, i.e. the locally constant functions. Indeed, a locally constant function such as $$f(x)=\begin{cases} 1, x<0\\ 0, x>0 \end{cases} $$ can be seen as the class represented by the family $\{\gamma _{(-\infty,0)}=1$, $\gamma_{(0,\infty)}=0\}$.
For an actual proof that the sheafification/completion of the constant presheaf is the sheaf of locally constant functions, see this or this. There, it is equivalently described as the sheaf of continuous functions to $G$, equipped with the discrete topology. You will, however, have to know at least the notion of stalk, which Spanier doesn't seem to introduce, from what I can tell.