As someone who grew up in a conservative household in a deep red state, I think that part of it is that a lot of people are letting Lizard Brain dictate their response to transgender people.
Let me give you a personal example. A while back, I went to a social dance, and there was a trans woman there. Before the dance starts proper, the couple that runs it will teach a dance lesson, and we rotate partners while that's going on. Eventually, I was rotated into being her partner. For some background, she was obviously early on in her transition; she still looked like a dude in a dress, she didn't quite have the appearance down yet. But she gets huge props for not only having the bravery to go out as herself, but doing it in fucking Arkansas.
So I rotate over to her, and it dawns on me that she's trans. In my head, Lizard Brain immediately starts screaming. "WHAT THE FUCK IS THIS?! THIS PERSON IS OBVIOUSLY A DUDE IN A DRESS, HE MUST BE UP TO SOMETHING IF HE'S DRESSING AS SOMETHING DIFFERENT THAN WHAT HE IS! RABBLERABBLERABBLERABBLERABBLE"
Keep in mind, where I grew up, you just didn't see trans people, and even now, it still tickles that primal part of my brain that was trained to be uncomfortable around people who aren't white and straight.
The difference between me and many of the people I grew up around is that I recognize that it's happening and try to tone Lizard Brain out when it starts screaming. A lot of other people listen to it and don't care that the person that it's screaming about is exactly that: a person.