He won Ginger Rogers over with his dancing in The Gay Divorcee. (Back when gay meant, well, gay, as in numbers one through four here or as in this song.).
As implausible as it would be for mere mortals to win a girl with their dancing, it totally made sense with Astaire. Astaire was amazing! Mikhail Baryshnikov, one of the greatest dancers of all, once put it this way: "No dancer can watch Fred Astaire and not know that we all should have been in another business.”