I very much agree with Flora and Basil.
TERF is used to get radical feminists and anyone who agrees with their analysis of trans/gender/sex to STFU. It is also used to 'other' rad fems so that people will be put off even hearing them out because they want to distance themselves from 'transphobia' because nobody likes to think of themselves as a bigot. This strategy is proving very successful.
It is used in a very similar way to that of how radical feminists are accused of being racist and excluding women of colour. Nobody wants to be thought a racist so better steer clear of those racist radfems (when in fact liberal feminism was no less neglectful of women of colour, indeed it was more neglectful. Radical feminism is very very definite on the tenet that ALL women count and ALL women matter and that includes poor women, women of colour, women with disabilities, older women, all women who traditionally bear the brunt of misogyny. This is one of the reasons that radical feminists are so against prostitution and pornography - because they are racist institutions which disproportionately harm women of colour. It is also one of the reasons for the radicalism in radical feminism - we do not believe that liberation for all women can be achieved within a racist, sexist, capitalist, edifice).
Of course the difference is that racism is very much alive in society and society is white supremacist. And white feminists, no matter how just and non-racist they may wish to be, are products of white supremacist society and therefore have white privilege and all the usual accompanying issues with entitlement, etc that we need to be mindful of.
So although TERF is used in a similar way to the accusation of racism, WRT to silencing radical feminists and women who agree with them, the issues of transphobia and racism are fundamentally different.
Having said that, accusations of radical feminism/feminists being racist, do very often come from white women - usually those that disagree with radical feminism and accusations of racism are a good way to discredit people you disagree with. I think this is similar to what almondcakes is saying about white middle class educated women jumping on the TERF/transphobia bandwagon. I don't think a lot of these women really care terribly about trans issues - I think they use trans issues in order to get radfems to STFU and in order to present themselves/think of themselves as jolly progressive and cutting edge (intersectionality can be used in this manner too).
As to why trans issues seem to take up so much space, well, Flora was bang on the money IMO. Trans issues take up so much space because men take up so much space and trans issues are extremely male centric. Plus the space they want is not that of other men (which would be much harder to muscle in on), the space they want is that of women. And everyone knows that women's job is to make sure the men are all right and to put women last because men matter and women don't. And men's issues are legitimate and women's issues are not.
I put transphobia into inverted commas, not because I don't think it exists but because I disagree with the definition it has taken on. Women not wanting transwomen, who have their penises intact, in our locker rooms and prisons are not being transphobic. Women who want to talk about (and have feminist discussions about, or meet up in women only safe spaces to exchange on) gynecological issues, pregnancy, menstruation, vaginal rape, FGM, abortion, PIV, hormonal contraception, etc. are not being transphobic. And it takes a particular kind of male privilege/sense of entitlement/narcissism to think that we are being transphobic WRT the above.