There's an important point being missed here, which is the etymological one.
Flamboyant is a French word, meaning "flaming." It's the present participle of the verb flamboyer, to be on fire. It has the same root as the cooking term flambé, which basically means "set-on-fire."
So at least originally, to be a flamboyant homosexual would be to be a flaming homosexual. I assume we've taken this term back from the bigots, so what I write here, I write without hate in my heart. To be flamboyant would mean being over the top, brightly colored, visible. Say, neon-colored vests covered in sequins, gamboling down the street, David Bowie-esque levels of grooming, that whole bag. (Mind, I'm not saying this is a bad thing. If I were gay, I'd like to think that I would be completely over the top. It looks fun!)
BookReader raises another interesting point, that the epithet faggot was, at one time, primarily used to refer to firewood (specifically kindling, if my knowledge of fire-making serves). So several of the inflammatory terms thrown around in reference to identifiably gay people have to do with fire. I would suggest that a very interesting node could be made analyzing this cultural nugget.
In a nutshell: the other writeups here are correct. But there's a very simple, etymological explanation of where this phrase came from.