How should we think about the sections of a sheaf on a scheme as functions? I have to admit it is a little embarrassing that I have come this far learning about schemes without feeling like I am really comfortable with this. Suppose $(X, \mathcal{O}_{X})$ is a scheme. In fact, for the purposes of this question it probably suffices to just consider an affine scheme.
What I am really wanting to know is, how should we $\textit{really}$ think about the sections of $\mathcal{O}_{X}(U)$? Or perhaps an equivalent question: How are the different formalisms equivalent?
To elaborate, consider the treatment in Eisenbud & Harris. There, they treat the "functions" on an affine scheme $\text{spec}A$ as being elements $f \in A$ whose action on a point $p \in \text{spec} A$ (corresponding to prime ideal $\mathfrak{p} \subseteq A$) is something like
$$   f: p \mapsto A / \mathfrak{p} \longrightarrow \kappa(p),  $$
with $\kappa(p)$ being the residue field at $p$. So in this sense, the sections can be thought of as functions mapping points into a different field at every point. 
Hartshorne begins simply by stating that the sections over $U$ are functions 
$$
s: U \longrightarrow \bigsqcup_{p \in U} A_{\mathfrak{p}}
$$
which satisy certain localness properties. This formalism of Hartshorne seems very analogous to the construction of the sheaf associated to a presheaf. Indeed that construction does exactly the same thing: You consider maps from the open sets into the disjoint union of stalks and impose certain localness conditions. So is the sheaf that Harthshorne defines for affine schemes somehow the sheafification on something much more simple?
So how exactly are all these ideas tied together? I feel like something is not really clicking. When we talk about a section of $\mathcal{O}_{X}(U)$, what is this object $\textit{really}$? As a function, what does it do? How does the collection of these functions give rise to the sections of bundles of germs as in Harthsorne? 
I've tagged this as a soft question since I'm not really sure it has a concrete answer beyond an intuitive explanation.
 A: Given a ring $A$ the structural  sheaf $\mathcal O_X$ of the affine scheme  $X=\operatorname {Spec}A$  is indeed the sheaf associated to a very natural presheaf, namely the presheaf of rings  $\mathcal O'_X$ defined as follows:   
For an open subset $U\subset X$, the ring $\mathcal O'_X(U)$ is the ring of fractions $$\mathcal O'_X(U)=S(U)^{-1}O_X(U)$$ where the multiplicative set $S(U)$ is $$S(U)=\{f\in A\vert\,\forall \mathfrak p\in U, f\notin \mathfrak p\}$$   You can find an example proving that in general $O'_X(U)\neq O_X(U)$ here.
Note that for $U$ of the form $U=D(f) \; (f\in A)$ we do have $O'_X(U)= O_X(U)$: Hartshorne Proposition 2.2 (b), page 71 in chapter II.
Edit: and what about differential manifolds?
It is an extremely amazing and underappreciated fact for an open subset $U\subset X$ of a differential manifold $X$ we do have $$C^\infty(U)=\{\frac {g\vert U}{f\vert U}: f,g\in C^\infty(X)\operatorname {and}\forall x\in U, f(x)\neq0 \}$$ 
So if  manifolds  are  seen as  locally ringed spaces (as they should!) we do have in that category $\mathcal O_{X,\operatorname {diff}}^{'}= \mathcal O_{X,\operatorname {diff}}$ !
One of the very rare references on this result is Nestruev, Proposition 10.7, page 145.
A: If $X$ is a scheme over a field, then we can really interpret a section as a function with values in the field.
Note that for any open $U \subset X$ a morphism $U \to \mathbb A^1_k = \operatorname{Spec}k[x]$ is the same as a map $k[x] \to \mathcal O_X(U)$ and by the universal property of the polynomial ring this is the same as a choice of an element $f \in \mathcal O_X(U)$. Thus we have natural bijection between section of the structure sheaf and morphisms to the affine line. If $k$ is algebraically closed, the points of the affine line can be interpreted as points in $k$.
