You know, I really don't think the majority of people are anti-trans. By which I mean I think most people really are happy to let other adults lead the life they want, and that they support laws and institutions treating trans people with exactly the same respect as anyone else.
There are, though, some things that I believe the majority of people object to or disagree with, which the trans community and allies might regard as anti-trans.
Firstly, most people don't agree with statements that 'trans women/men are women/men'. To be clear, they agree they should be treated as such if they wish, but not that they actually are those things.
The words 'woman' and 'man' have existing meanings and the majority have spent their lives using those words according to those meanings. You can see this is true when you look at dictionaries which record the use of language. It's annoying to many that a small group of people has decided on behalf of everyone else that those meanings now do not apply. I personally think it was a huge mistake of the trans community to make agreeing with these statements a test for transphobia. Most people, if they are honest, don't agree with them.
By the way, I don't think this means its right to insist on calling call a trans woman a man. That's simply rude and probably does suggest you are transphobic.
Secondly, many women, in particular, see their biology as a very important element in their identity as a woman. A patriarchal world has reacted to them, often in hostile or unwelcome ways, because of their biology. It has judged them for being pretty or not. For having boobs, or not. It has threatened their safety because they can be the object of sexual desire from men, and it has spurned them when they are not. It has paid them less because they have the potential to bear children, and has locked them into drudgery when they do. Women have suffered these things not because they identify as women, but because their biology makes them women, so it pisses them off to be told their biology doesn't matter.
Finally, the majority see trans women in competitive sport - above a very junior, non-competitive level at least - as being unfair. They see this as so obvious that it barely needs further explanation, and reject excuses for it as gaslighting.