As a young woman I was definitely seduced by Rand: so I can offer a few ideas. I succumbed due to "The Fountainhead."
1) The "rape" in Fountainhead was very ambivalently presented, but this I can say about it: it was the only time (outside of Chaucer) that I ever read an account in which a woman has sex, enjoys it, and does not get punished for it. The permutations of this are infinite; but for some reason women do not get to come in western lit and then go on to have happy lives. My mother had been very explicit about the plumbing of sex, but she never suggested that there was anything enjoyable about it. So, when, at the age of 12, I read Fountainhead and got to the passage where Dominique has sex and then does not want to wash it off, I had a real revelation.
2) Rand's characters are all extremely egocentric and "self-made" -- a favorite teenage fantasy. (It follows, of course, that there are no children in her universe...just very small, already formed adults.) Another reason for her teenage appeal.
3) Rand's villains are all fake philanthropists: they get their power by pretending to be selfless, but they are actually far more manipulating and dangerous then the people who are consciously selfish. This also appeals to teenagers who are completely guilted out by their "selfless" parents: "We disrupted our lives to bring you to this great free country and you are an ungrateful wretch..."
So basically, unless you outgrow your adolescence, you'll worship Rand all your live long days.
I don't think it goes much deeper than this, except has anyone noticed what an insane Stakhanovite she is? Her heroic characters work around the clock! Also appealing in a society in which no social relations are possible.
Joanna