I think Kanigel's book talks about this. It seems he had worked through first few thousand numbers and knew all of them quite well. Thus instead of a spark of genius, this seems to be a fruit of years worth of hard work and an excellent memory.
Few thousand? There are only 12 third powers below 1729: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728.
Spotting that 729 + 1000 almost is equal to 1728 is not hard, either. Everybody who has computed that list will have noticed it. Translating that observation into a concise, interesting description is not hard, either, but it does require creativity. That, he had way more than most. I wonder what he would have said about the pair 3^5 and 7^3 (243 and 343)
He was a genius,Yet shy person ,Its said that he also gave properties of many other numbers which were not documented by anyone, But only Hardy knew them and would tell it to others.
http://en.wikipedia.org/wiki/1729_(number)