I think a lot of people genuinely believe that transwomen are grooming and raping children en masse, or at least trying to get into women's sport so they can win medals. The fact that this is both wrong and stupid and has no real non-circumstantial evidence does not stop them.

At some point people have to talk to each other, right? And that's where you have a discussion about how you don't think kids should go to the Folsom Street Fair either, but that you also don't think it's fair for a 39 year old transwoman not to be allowed onto the division F basketball team with her friends, and maybe you and your interlocutor both discover that the other side is a little more tolerable than you thought.

Edit: I didn't really explain that very well. What I mean is that neither extreme finds the opinions of the other extreme tolerable, and that this is the result of paying too much attention to the long tail extremes instead of the middle of the bell curve.