I'm sure this isn't a complete answer, but from what I've seen from TERF rhetoric, it seems to be basically "girls rule, boys drool". They need to believe that men and women are different so that they can easily split humanity into a very basic us vs them dichotomy, with themselves in the "good" group. I don't know why they want to make sex/gender the fundamental contradiction in their lives, it seems to me there are way more explanatory ways of looking at the world, but every single TERF argument seems to boil down to hating people they perceive as men and thinking the people they hate are biologically programmed to be evil.
But then they rarely take this idea to its logical conclusion, which would be cis-female only enclaves without any cis men or trans people present at all. Even the most rhetorically strident TERFs tend to have a cis man or two in their life who they don't seem to hate. And I have to admit I think that's odd. If someone truly in their heart of hearts believed that evil lived in the penis (or XY chromosomes or whatever), you'd think they would try harder to avoid people with those traits. You'd think genital inspections and maybe even chromosome tests would be a prerequisite to friendship with someone with that (incredibly shitty) belief.
I dunno, I do think hatred and disgust are at the root of TERF beliefs, but I don't think most TERFs really interrogate their own beliefs very much.
I don't think Julia Serano is totally and completely right about that. She's great, and Whipping Girl is great, very worth reading, it really helped me flesh out my understanding of gender, but still, I think she's wrong about, at least, the subconscious sex theory she posits.
I'd like to recommend Judith Butler's new book here, but I'll be honest, I can't read Butler, I find them extremely confusing. For an easier time, you could instead watch Philosophy Tube's newest video, because Abby talks a little about this issue, following Butler. If you're interested, I also could see if my sister would be ok with me posting an excerpt of an essay she wrote on this subject, because that's my exposure to Judith Butler, filtered through my sister's writing.