That must be what a mathematician feels when they see a beautiful proof, or formula, or relation.
Statements that are obvious usually aren't interesting and aren't beautiful. (though there may be some like e^(i*pi) = 1 which are obvious consequences of the definition of the complex exponential function that people seem to like -- to me I think the definition of complex exponentiation is the beautiful part) The result in some sense should be counter-intuitive -- it had to have taken a spark of genius to see it.
Once you see it, it should seem confusing at first, but the proof of it should cause you to say "of course, this is the only way it could have been."
Finally, once you know about it, you immediately see useful consequences of it that open all new lanes of inquiry, or practical applications for it.
edit: I think also "simplicity" is a core part of it in that it distills a complex set of ideas to a concise statement -- e=mc^2, for example, or the example above.
There is a lot going on for summing 3 squares. It is fun and unexpected. All the weirder because they don't seem to have linked all this to pi, e or i yet in an obvious way looking at the wiki page.
It also takes a mighty big dog to weigh a ton.