Emma Allen of Radical Women attempts to explain the transphobia exhibited in some parts of radical feminism:
At the heart of the attacks on transgender people is the traditional radical feminist notion of biological determinism, which interprets humans and human life from a strictly biological point of view — holding that biology is destiny. Their view that women’s inferiority is based on their biology and that men are the enemy, is a reverse image of patriarchal hatred of women. The basis of radical feminism is to see men as the problem, painting women as the natural victims of men. If women are oppressed specifically because of the reproductive organs they are born with, rather than a deeper social-economic source of gender inequality, then transwomen can’t be part of the club. Accepting the sisterhood of non-biological females challenges the very basis of radical feminism.
The Radical Feminism talked about here is that current in feminism that sees the patriarchy, the systemic oppression of women by men as the root of all oppression, privileging it above race, class or sexuality based oppression. In the socalled Trans-Exclusionary Radical Feminism version of this, this belief has hardened into a belief that gender is exclusively biological in nature, that men and women are in opposition and hence therefore any trans woman is nothing but at best a spy, an intruder. At the same time because, as Allen explains, this current of Radical Feminism also believes that the feminist revolution can only be completed if gender is abolished entirely as a concept, trans women are a direct threat to their ideology, as obviously they show gender goes deeper than the gender expression radical feminism recognises.
Emma Allen’s own form of feminism, socialist feminism on the other hand recognises that:
In contrast to radical feminists, socialist feminists view the private property system as the historical and economic foundation for patriarchy and the subordination of women and sexual and gender outlaws.
[…]
The role capitalist society has assigned to women is directly challenged by the existence of transgender, lesbian, gay, bi-sexual, intersex and queer people – which is a good thing!
Unfortunately however Trans-Exclusionary Radical Feminism is more prominent a voice in radical feminism at the moment. Born out of second wave feminism, many of its adherents and allies like Julie Birchill, Germaine Greer or Sheila Jeffreys have a voice through the mainstream media less accessible to trans feminists and their allies. And as Tina Vasquez lays out in Bitch Magazine, TERF feminists use their influence to attack and hurt trans women:
For example, transgender people were able to readily obtain government-funded healthcare prior to 1980. That year, Janice Raymond wrote a report for the Reagan administration called “Technology on the Social and Ethical Aspects of Transsexual Surgery” which informed the official federal position on medical care for transgender people. The paper’s conclusion reads, “The elimination of transsexualism is not best achieved by legislation prohibiting transsexual treatment and surgery, but rather by legislation that limits it and by other legislation that lessens the support given to sex-role stereotyping.” In her book Transgender History, Susan Stryker says that the government curtailed transgender access to government social services under Reagan, “In part in response to anti-transgender feminist arguments that dovetailed with conservative politics.”
This is why Trans-Exclusionary Radical Feminism matters.