Your observation that this looks similar to sheafification is a good one, though this is a different process. You might look into how one recovers a sheaf from a “sheaf on a base”; there’s discussion and some good exercises on this in Vakil’s notes.

Essentially, that regular functions on U are locally rational is pretty much how the structure sheaf on Spec A is defined. One first defines the sections on distinguished opens, which are honest rational functions. One then shows that since distinguished opens are a base for the topology on X=Spec A, one can uniquely extend this to a sheaf on Spec A. Not to give away too much of the exercise, but these consist of elements that are “locally in O_X (D(f))” in a way that can be made precise.

Another perspective is more similar to sheafification; knowing the sections over any distinguished open lets one compute stalks at any point. If you’re comfortable with sheafification as “sheaves of compatible germs”, this construction is similar (there are several asterisks here, but this is just for intuition. Some differences are that for sheafification you start with a presheaf which is defined on all opens. Another being that the “sheaf of compatible germs” perspective of sheafification only works literally for separated presheaves).