Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category of open sets on $X$ to the category of rings, assigning to each open set $D_{f} \subseteq X$ a local ring $\mathscr{O}_{X}(D_{f}) = R_{f}$ and to each inclusion $D_{f} \subseteq D_{g}$ a ring morphism $R_{g} \rightarrow R_{f}$.
It is clear that $\mathscr{O}_{X}$ is indeed a presheaf on $X$, but I'm having trouble seeing why it is a sheaf on $X$, particularly the second sheaf axiom (gluing axiom). The axiom says that for any open covering $\bigcup U_{i} = U \subseteq X$, if for any section $f_{i} \in \mathscr{O}_{X}(U_{i})$ such that for all $U_{i}, U_{j} \subseteq X$ we have that $f_{i}|_{U_{i} \cap U_{j}} = f_{j}|_{U_{i} \cap U_{j}}$, then there is some section $f \in \mathscr{O}_{X}(U)$ such that $f|_{U_{i}} = f_{i}$ for all $i$.
How is this explicitly shown to be true for $\mathscr{O}_{X}$? I have attempted (rather poorly) to start naturally with the assumption that $f(\mathfrak{p}) = g(\mathfrak{p})$ for all $\mathfrak{p} \in U_{i} \cap U_{j}$, where $f$ and $g$ are defined on $U_{i}$ and $U_{j}$ respectively. But what function in $\mathscr{O}_{X}(U)$ equals $f$ or $g$ when considered in the restrictions $\mathscr{O}_{X}(U_{i})$ and $\mathscr{O}_{X}(U_{j})$?
Strictly speaking the association of $D_f$ to $R_f$ only defines the sections on a basis; to construct the regular functions on an arbitrary open $U$, we have to take the inverse limit of the $D_f$ contained in $U$. The fact that this is a contravariant functor is pretty straightforward from the universal property of inverse limits. Separatedness is also fairly easy. Recall that on a module 'being equal to 0' is a local property, or explicitly for $x \in M$ we have $x=0$ iff $x_{\mathfrak{p}} = 0$ for all prime $\mathfrak{p}$. Separatedness is the geometric dual of this fact. The sheaf axiom is mor involved, but a good algebraic proof can be found in 'The Geometry of Schemes'.
Personally I think that sections of a scheme are best thought of as stalk-valued functions on $U$. In this view all the sheaf axioms are trivial and its simple to make the connection between this etale definition and the definition involving principal opens.