$\begingroup$

I'm not a number theorist, but FWIW: I would talk, not so much about Gödel's Theorem itself, but about the wider phenomenon that Gödel's Theorem was pointing to, although the terminology didn't yet exist when the theorem was published in 1931. Namely, number theory is already a universal computer. Or more precisely: when we ask whether a given equation has an integer solution, that's already equivalent to asking whether an arbitrary computer program halts. (A strong form of that statement, where the equations need to be polynomial Diophantine equations, was Hilbert's Tenth Problem, and was only proven by the famous MRDP Theorem in 1970. But a weaker form of the statement is contained in Gödel's Theorem itself.)

Once you understand that, and you also understand what arbitrary computer programs can do, I think it's no surprise that number theory would seem to contain unlimited amounts of complexity. The real surprise, of course, is that "simple" number theory questions, like Fermat's Last Theorem or the Goldbach Conjecture, can already show so much of the complexity, that one already sees it with questions that I intend to explain to my daughter before she's nine. This is the number-theoretic counterpart of the well-known phenomenon that short computer programs already give rise to absurdly complicated behavior.

(As an example, there are 5-state Turing machines, with a single tape and a binary alphabet, for which no one has yet proved whether they halt or not, when run on a tape that's initially all zeroes. This is equivalent to saying that we don't yet know the value of the fifth Busy Beaver number.)

Here, I think a crucial role is played by a selection effect. Above, I didn't talk about the overwhelming majority of 5-state Turing machines for which we do know whether they halt, nor did I talk about 10,000-state TMs---because those wouldn't have made my point. Likewise, the number-theory questions that become famous, are overwhelmingly skewed toward those that are easiest to state yet hardest to solve. So it's enough for some such questions to exist---or more precisely, for some to exist that are discoverable by humans---to give rise to what you're asking about.

Another way to look at it is that number theorists, in the course of their work, are naturally going to be pushed toward the "complexity frontier"---as one example, to the most complicated types of Diophantine equations about which they can still make deep and nontrivial statements, and aren't completely in Gödel/Turing swampland. E.g., my layperson's caricature is that linear and then quadratic Diophantine equations were understood quite some time ago, so then next up are the cubic ones, which are the kind that give rise to elliptic curves, which are of course where number theory still expends much of its effort today. Meanwhile, we know that if you go up to sufficiently higher complexity---apparently, a degree-4 equation in 9 unknowns suffices; it's not known whether that's optimal---then you've already entered the terrain of the MRDP Theorem and hence Gödel-undecidability (at least for arbitrary equations of that type).

In summary, if there is a borderland between triviality and undecidability, where questions can still be resolved but only by spending centuries to develop profound theoretical machinery, then number theory seems to have a pretty natural mechanism that would cause it to end up there!

(One sees something similar in low-dimensional topology: classification of 2-manifolds is classical; 4-manifold homeomorphism is known to be at least as hard as the halting problem; so then that leaves classification of 3-manifolds, which was achieved by Perelman's proof of geometrization I've since learned this is still open, although geometrization does lead to a decision procedure for 3-manifold homeomorphism.)

In some sense I agree with Gerhard Paseman's answer, except that I think that Wolfram came several generations too late to be credited for the basic insight, and that there's too much wrong with A New Kind of Science for it to be recommended without strong warnings. The pictures of cellular automata are fun, though, and do help to make the point about just how few steps you need to take through the space of rule-systems before you're already at the edge of the abyss.