Technically, the following are not proofs, or even theorems, but I think they count as insights that have the quality that it's hard to imagine computers coming up with them. First, there's:
Mathematics can be formalized.
Along the same lines, there's:
Computability can be formalized.
If you insist on examples of proofs then maybe I'd be forced to cite the proof of Goedel's incompleteness theorem or of the undecidability of the halting problem, but to me the most difficult step in these achievements was the initial daring idea that one could even formulate a mathematically satisfactory definition of something as amorphous as "mathematics" or "computability." For example, one might argue that the key step in Turing's proof was diagonalization, but in fact diagonalization was a major reason that Goedel thought one couldn't come up with an "absolute" definition of computability.
Nowadays we are so used to thinking of mathematics as something that can be put on a uniform axiomatic foundation, and of computers as a part of the landscape, that we can forget how radical these insights were. In fact, I might argue that your entire question presupposes them. Would computers have come up with these insights if humans had not imagined that computers were possible and built them in the first place? Less facetiously, the idea that mathematics is a formally defined space in which a machine can search systematically clearly presupposes that mathematics can be formalized.
More generally, I'm wondering if you should expand your question to include concepts (or definitions) and not just proofs?
Edit. Just in case it wasn't clear, I believe that the above insights have fundamentally changed mathematicians' conception of what mathematics is, and as such I would argue that they are stronger examples of what you asked for than any specific proof of a specific theorem can be.