This isn't comprehensive, but I'll do my best.
TERFs are, first and foremost, radical feminists. Radical feminism is essentially second-wave feminism without the intersectionality brought in by third-wave feminism. It believes that patriarchy is at fault for the oppression of women, but sees this in a very strict, binary way: women are the oppressed, and men are the oppressors.
TERFs use this to justify their specific brand of transphobia. This idea, among others, is essential in supporting that transphobia.
I'll try to outline some of those ideas, and some of the logical thruoughlines they use:
Women are uniquely oppressed, and always in danger. Womanhood- or the experience of being a woman- is defined by oppression, misogyny, and Being In Danger.
Women are particularly in danger in the presence of, and in relationships with, men. Spaces that exclude men are essential to preserving the safety of women.
Socialization: men are raised to support patriarchy, while women are raised to be subjugated by it. Men have no motive to unlearn these lessons, so all men are inherently more corrupted by these lessons than women.
Relationships with men are therefore inherently (more likely to be) abusive, and relationships with women are inherently safe(er).
Sex, in particular, is more often exploitative than not. Only some kinds of sex are not exploitative. Many kinds of sex that we think are consensual, or that people say are consensual, are either rape or proto-rape.
Exchanging money for sex is inherently rape/exploitation/non-consensual in some way.
As women who deny men access to them, lesbians are The Most Oppressed and also The Most Endangered. They must be protected at all costs.
Because so many women have been raped by men with penises, both men and penises are inherently traumatic to A Lot Of Women.
Many lesbians will naturally have an aversion to relationships with trans women because of this. Trans women who argue against this "genital preference" are potential rapists trying to infiltrate lesbian spaces to hurt and take advantage of women.
Men will always try to invade "women's spaces" to take advantage of women, endanger them, and strip away their resources both for personal gain/pleasure, and in service of upholding the patriarchy.
If we allow men to say they are women, they will invade those spaces and hurt "real" women. Men who say they are women are dangerous, and must be excluded and punished.
Men may try to obfuscate labels and terminology to "define women out of existence" or otherwise cause confusion, which they can manipulate to further their infiltration.
Women are all miserable with their bodies, cursed with the pressure to reproduce and have sex with men.
Women are all miserable with their genders, forced as they are to ensure the overwhelming and constant suffering that is patriarchy.
Women will attempt to escape this misery and pressure by "becoming men". This is cowardly, but understandable; a tragic but inevitable result of patriarchy. These women must be saved.
Some women who try to escape patriarchy are doing it out of self-interest; they are betraying women by becoming men, and contributing to their oppression. These women must be punished.
Bio-essentialism: women are oppressed specifically because of their bodies and ability to reproduce. This is an inherent and defining part of womanhood. Nobody can claim womanhood without this experience, everyone who has had this experience is a woman.
Women's bodies are all beautiful and perfect because they are women's bodies. If the womanliness of them is tampered with, they become less valuable. Men's bodies are gross and undesirable symbols of patriarchy.
Testosterone makes people violent, aggressive, irrational, and angry. Estrogen makes people calm, kind, and happy.
Men can never understand women's bodies as well as other women do.
People can be attracted to other people on the basis of "sex" alone. This is inherent, immutable, and unquestionable.
Men are sexual animals who inherently and unavoidably find lots of bad things sexually arousing. Because "youth" is attractive, many men find young girls and children attractive, and will try to take advantage of them. Misogynistic control/power over women, hurting women, and even rape are also inherently sexually appealing to men.
"Gender" is meaningless; it's founded in misogynistic stereotypes about men and women, and when you remove the stereotypes, there's nothing left at all. Only binary "sex" is real, because that's what patriarchy (and biology) is based on.
Manhood is itself a toxic, oppressive, inherently corrupting concept. Anyone who participates in manhood is corrupt and immoral; who would choose to be the oppressor?
Masculinity is defined only by hating women, having power, and being aggressive, violent, and controlling (etc.)
Patriarchy doesn't just target women, but femininity as a whole, for its association with women.
Patriarchy doesn't just reward men, but masculinity, as it rejects femininity. People who reject femininity and embrace masculinity are rewarded by the patriarchy.
Some of these ideas are contradictory, but they lead to the same conclusions. Some of them lead to similar conclusions, many of which take very little further nudging to push into more dogmatic ideas.
This is exactly why we need to understand all of these paths into TERF ideology- and more.
In fact, the vast majority of the points on this list- particularly the beginnings of their logic- can be very easily swallowed while still holding that trans women are women, and trans men are men.
That's what TIRFs (trans-inclusive radical feminists) are, and they're still incredibly dangerous. TIRF ideology normalizes these points, making it far easier for TERFs to recruit; even if TIRFs themselves try to be aggressively anti-TERF.
Again, this isn't comprehensive, and it would take a long time and a lot of words to cover every flaw and danger in every line of reasoning here.
But remember how these things work; even if some of them begin with a grain of truth, even if some of them are true- especially if you define the words they contain differently- be wary of them.
