To them, it's as if a man were allowed to enter the Miss America pageant. They think two things:
1. "Why do you even want to be here? Are you trolling us?"
2. "When men are allowed to enter the Miss America pageant, identifying as a woman becomes a farce because all the characteristics we've associated with being a woman have been rendered invalid."
So it's not that they are afraid their rights are being infringed. It just makes them feel like their own marriage is devalued if the definition of marriage is expanded to the point where it becomes meaningless.