When I first learned algebraic geometry a few years back, I was taught that for an affine variety $X \subset \mathbb{A}^n$, a function $f$ was regular at a point $p$ if there existed a neighbourhood of $p$ where one could express $f=\frac{g}{h}$, $g,h \in \mathbb{F}[x_1 , \dots , x_n]$, $h \neq 0$. The ring of all functions that were regular at every point on $X$ was just called, plainly, the ring of regular functions, denoted $\mathcal{O}(X)$, and by a nice proof, it emerges that this was in fact isomorphic to the coordinate ring of $X$, $\mathcal{O}(X) \cong A(X)$.
Something about this all always kind of bothered me. While I appreciated the usefulness of isomorphism, and I could abstractly understand every step of the proof (which was the one in Hartshorne's book, for the record), I could just never understand how the proof could ever have come about, as in, who would ever have guessed that these rings of "fractions of polynomials" would be something isomorphic to the coordinate rings? Why would you even begin to investigate that? Where was the link where, so to speak, you could see that, "Oh, clearly, there's clearly correspondence between functions $f(x) + I(X)$ and functions that can be expressed $\frac{g}{h}$" to provide motivation for the definition of a regular function, and then the proof in question.
And so, basically, that's my question: what's the clear, initial observation between the relationship of functions $f(x) + I(X) \in A(X)$ and functions that can be expressed $\frac{g}{h}$, $g,h \in \mathbb{F}[x_1, \dots , x_n]$.
If the question is a bit too fuzzy, please let me know, and I shall try to clarify.