Why is the group of sections of a sheaf of modules given by the group of morphisms of sheaves?

Question

Let $(X, \mathcal{O}_{X})$ be a ringed space and let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X}$-modules. Why is it true that $$ \text{Hom}_{\mathcal{O}_{X}|_{U}} \left( \mathcal{O}_{X}|_{U} , \mathcal{F}|_{U} \right) = \mathcal{F}(U)? $$ I feel like this should be obvious, but I am not seeing what the isomorphism is. How does a family of maps on the groups of sections for subsets $V \subseteq U$ correspond to an element of the module $\mathcal{F}(U)$?