I think most mathematical facts described as beautiful fall into one of the two categories of “consequences of definitions” and “shocking connections”. The first happens when the structure of your terms is lined up in just the right way as to make a proof feel automatic and clear, every piece follows right from the previous one in a natural way. The second one is rarer imo, and is enjoyable almost in the same way a clever punchline is. A series of facts are setup, and then your viewpoint is suddenly shifted forcing you to recontextualize those facts and see something new. A lot of “elegant” proofs are of this flavor.