"Why, exactly, do you need to separate trans women from women for women to have rights?"
Firstly, women are an oppressed group of people. And they are oppressed by men. Women's lack of rights is caused by male oppression. Recognising these hierarchical structures and working to combat them is important for any civil rights movement (see black liberation, gay liberation, etc.)
There has never been a rights movement, however, where the oppressing class has declared itself to BE the class it is oppressing, and to even say that it is more oppressed. Except for now, with women's rights. Despite being oppressed by men for millennia (and often on the basis of our biology), women are now being told by males that they are women, because woman now has nothing to do with bodies or being female but is now entirely a mental construct which can instantiate inside a male's brain. Woman is a thought. And men can be women by thinking. By virtue of a mental state, men can jump from the oppressor class to the oppressed class.
If the distinction between men and women is broken down, if the oppressor can become the oppressed just by thinking it and declaring it to be so, then it is hard to see how there can even be 'women's rights' because there simply isn't such a thing as 'woman' that is a meaningful category other than "a person who is apt to declare themselves as such."
Secondly, women have special rights and needs by virtue of our female body. Only females can give birth, can get abortions, have periods, have female reproductive cancers, etc. Our female bodies are not catered to by science or medicine or technology (see the book 'invisible women'. ) Our bodies are highly sexualised and objectified by male society. The female sex has the right to gather away from males, in sports, domestic abuse shelters, hospitals, prisons, etc.
Transwomen, however, insist that they are female too and that they get access to these spaces as well. If transwomen gain access to these spaces, then women lose the right to female only spaces because by definition they are not female-only any more. So, in order for women to have certain rights it is essential that transwomen are kept separate in certain domains. This is exactly what the Equality Act recognises.