If we strip marriage of the asymmetry of power, as we must if we allow men to marry men and women to marry women, then we also strip away the man and wife, dominant and submissive, owner and owned, master and servant relationship that characterizes the conservative view of marriage.
Of course! I was always really puzzled how anyone could assert with a straight face that allowing more people to get married was somehow destructive towards the "institution of traditional marriage". But my confusion was a result of me thinking that the definition of "traditional marriage" was simply that it was between a man and a woman. If that were the case, then I would be right, same-sex marriage poses no threat to that "institution" because, with the possible exception of a few sexless sham marriages, the number of people getting "traditional marriages" would be exactly the same as before, and the couples in them would be entirely unaffected.
But if the definition of "traditional marriage" inherently implies patriarchy as well, then of course same-sex marriage is corrosive to that institution. No heterosexual person is going to look at someone else's same-sex marriage and say, "Hey, that looks like fun, I think I'm going to decide to be gay!": sexual orientation doesn't work like that. But it's entirely plausible that many heterosexual women might look at a happily married well-adjusted same-sex couple and say, "Hey, how come they treat each other like equals? Shouldn't I be treated as an equal with my husband?" And that's just bad for business. If you're a misogynist patriarchal theocrat, that is.