Anyone supporting the notion that women are not allowed boundaries against natal males without being hateful, and gay people are not allowed to be same-sex attracted but must also accept a person's subjective feelings about their mental gender instead, is supporting homophobia and misogyny. It's not transphobic to recognise biological sex is real, and matters, and affects women and gay people especially. On the contrary; it's bigotry to deny the validity of biological sex, because you also deny women's worldwide, millennia-long oppression, and that same sex attraction is a reasonable and valid orientation.
Women's oppression doesn't arise because we identify into a gender. It's based upon biological sex. Denial of that, and creating a new system whereby gender is a choice, removes the ability to name, and therefore challenge, sex-based oppression. And that is wrong. It's silencing, abusive, and breathtakingly effective. Men's Rights Activists must be open-mouthed in admiration.
As for, "why does their website mention trans people?" Well, funnily enough, feminist websites often mention men. If you're creating an org. to protect the interests of one group, threatened by another, you need to explain how, and why the other group's interests are being privileged against your own, and why that's a problem. It doesn't equal hatred of the other group - there are male allies to feminism, and trans people supporting both Women's Place and the LGB Alliance.
I can only imagine anyone posing that question agrees that all feminists must hate men, because they point out that sex-based oppression is a real thing that exists. Then again, perhaps they're too busy insisting feminists hate trans people to invoke the men one, these days.