Let $\mathscr C$ be a site, and $\mathscr T$ be its topos. $F \in \mathscr T, e$ the terminal object in $\mathscr T$.
We have the Yoneda embedding $T \mapsto \widetilde T = Hom(\cdot, T), T \in \mathscr C$, with a canonical identification $F(T) \cong Hom(T,F)$. In our case (crystalline site & topos), $\widetilde T$ is a sheaf.
According to "Notes on crystalline cohomology" -- Berthelot & Ogus, p. 5.15,
a section $s \in \Gamma(e,F) = Hom(e,F)$ is just a compatible collection of sections $s_T \in F(T)$ for every $T \in \mathscr C$, i.e. an element of $\varprojlim_{T\in \mathscr C}F(T)$.
By definition, clearly an $s \in Hom(e,F)$ gives $s_T \in Hom(T,F) \forall T \in \mathscr C$. However, I am not sure about the opposite direction. Did I misunderstand the statement?
EDIT: Sorry about my earlier misreading. It is still true that the terminal object in any sheaf category represents the limit functor. The point is that an element of the limit of a sheaf $F$ is, by definition, a cone over $F$ tipped by the terminal set. To give such a cone is to give an element of each $F(T)$, compatible with transition maps $T\to T'$; but this is precisely the definition of a natural transformation from the terminal sheaf to $F$. In other words, a compatible family of elements of $x_T\in F(T)$ uniquely defines a map $e\to F$ sending the unique section of $e$ over $T$ to $x_T$.