Take an elliptic curve. At each prime p you have some local information. It either has bad reduction, and there is a reduction type, or good reduction, and then you count the points and the number is $1+p-a_p$ with $a_p$ an integer between $-2 \sqrt{p}$ and $2 \sqrt{p}$. Also if the reduction is multiplicative, one should remember split or nonsplit reduction. One should also ignore the difference between reduction types that can be isogenous to each other. This information turns out to be the natural information about the elliptic curve to use in a whole lot of arithmetic situations.
Suppose I give you, for each prime, something that looks like this bit of information about the reduction of the elliptic curve - either a fiber type or a number $a_p$. How can you ever hope to tell if there is an elliptic curve that stitches all these disparate numbers together? Well, you can write down some necessary conditions, like that there are only finitely many primes of bad reduction. But there will still be only uncountably many "plausible" sequences, of which only countably many come from an elliptic curve. What makes them special?
Now take a modular form. This is an object that has primes of good reduction and bad reduction. The good reduction primes will just be the primes not dividing the level. At these primes, we have a Hecke eigenvalue, which is a real number between $-2 \sqrt{p}$ and $2\sqrt{p}$ - in fact it lies in the ring of integers of some number field, depending on the modular form. What's more, we can look at the action of the automorphism group of the modular curve on our modular form. If we look at the different representations that can appear, they turn out to be in perfect correspondence with the different reduction types of an elliptic curve (at least when the coefficient field is $\mathbb Q$).
So you have two different ways of stitching together this local information to get global objects - elliptic curves and modular forms whose coefficient field happens to be $\mathbb Q$. In both, you choose only a countable set of special sequences out of an uncountable set of possibilities. Could they be the same?
That would be quite impressive, because the ways they arise are so different! One comes from looking arithmetically at a natural class of algebraic geometry objects, and the other from doing analysis on some special symmetric spaces.
It would be particularly impressive because some facts (like the bound on the $a_p$) are much easier to prove on the elliptic curve side, while others (like a bound on the average of $a_p/\sqrt{p}$) are much easier to prove on the modular curve side.
Of course this sounds totally ridiculous. Those two things should be totally unrelated! Except that you can get, by a not-too-difficult construction, elliptic curves from modular forms...
So you have two ways to solve this incredibly mysterious problem of patching together local data at each prime to get a single global picture. They behave the same, and one is a subset of the other. Might the subset in fact be the whole - could these two pictures be the same? Once you came up with the question, and understood its importance, you would start searching for evidence, counterexamples, etc., and like the mathematical community in the mid-late 20th century, be convinced that they are.
Now this correspondence between elliptic curves and modular forms does not mention the actual map from the modular curve to the elliptic curve. Constructing it, giving the equality of $a_p$s, is nontrivial (the Tate conjecture for morphisms of abelian varieties) and doesn't really generalize. That is one reason you shouldn't think about this too geometrically. This is an arithmetic statement, not a geometric one. Note that it is only true for elliptic curves over $\mathbb Q$.
Also note that the fact that the modular curves are moduli spaces of elliptic curves is a coincidence, akin to the small number coincidences from Lie groups, like the fact that $PGL_2=SO_3$.
