NORMAL BEHAVIOR DEP'T.
"Why? Why would you go around thinking or caring at all about what other people have between their legs, if you're not personally having sex with them?": I expect you know the answers as well as I do, but here are two:
1. These people are unusually anxious and confused about sex. Many of them have been taught from an early age (as I was by the fundies who raised me) that sex is dangerous and shameful, so it must be tightly controlled. Any "deviance" at all undermines that control; it's the thin end of a wedge leading to who knows what debaucheries. (For fundies, relax an iota, and the next thing you know, you're Jerry Falwell Jr. and his wife having threesomes with the pool boy!) So the mere existence of transsexual people rattles their mental cages.*
2. These people are strongly committed to the prevailing status system, in which men are dominant and women are subordinate. They believe or want to believe this scheme is natural or divinely ordained and hence beyond legitimate challenge. Any blurring of the distinctions between men and women is a threat, most obviously to men who have little going for them except being men but also to women who are highly invested in their roles (as my mother was). Especially threatening to such men are people they construe as feminized men, whom they regard as, in effect, class traitors.
The syndrome is a nasty tangle of psychopathologies, which are not only undiagnosed and untreated but eagerly cultivated by demagogues for power and profit.
*Admittedly, this psychopathology isn't limited to fundies and their ilk. As many thoughtful observers have noted, many people feel deeply threatened by the fact that humans, particularly themselves, are mortal, breeding animals. Sex organs and sexual behaviors, which humans conspicuously share with other animals, are pointedly embarrassing reminders of this troubling fact. Accordingly, they're heavily obfuscated and regulated, including claims that maleness and femaleness are sacred categories that transcend corporeality. Even imaginary spiritual beings, gods and goddesses, typically are depicted as conventionally male or female, not anything other or in between. The messy biological realities of sex cast doubt on all that silliness, which may be why many people prefer not to think about them.