Hostility to trans people is a relic of second wave feminism which has clung on until the present day and gained popular traction through a number of arguments that are largely not backed by hard evidence and mirror those already used to hold back progress of marginalised groups in the past. Often these arguments rely on dog whistle politics (if you ever see anyone arguing that they stand up for “women’s sex-based rights”, this essentially means they are only concerned with pushing trans people back into the margins of society in the name of “protecting women.”)
The arguments themselves are a complete bag of contradictions. For example, they protest the idea of sharing women’s spaces with trans women yet to access hormones or surgery, whilst also simultaneously working to make it harder for trans women to access that care. Trans women have been using women’s spaces for years essentially entirely without issue - a few isolated examples of issues exist and are held up by TERFs as exemplary. Their arguments are invented to justify the inherent disgust that people feel towards trans people... I describe it as the same as that feeling homophobes get when they see two men kissing. Of course, it isn’t persuasive enough to just say that, hence the window dressing of women’s and children’s rights. The goal is literally to stop trans people from existing (which of course can't really happen, but in practice from having information that helps them understand their gender identity, feeling safe to come out, from being able to access gender affirming care, etc)
It started off with a bedrock of insane TERFs opposed to reform of the Gender Recognition Act, which dictates how trans people receive legal recognition for their gender through a currently very expensive, humiliating and laborious process. Their voices have been amplified by cash from the Christian Right and a media who are fond of scapegoating minority groups to distract from real problems. The reason it seems to be such a problem in the UK is that TERFism has caught on across the political spectrum, whereas even in the US, transphobia appears to be largely the preserve of the right. The family values angle does brings in Conservatives here, but in the UK the feminist angle also appeals to a sizeable chunk of leftists and liberals. I'd say middle aged and middle class women are the absolute bedrock of the movement and their fervour is almost cult-like in nature.
I'm very much happy to answer anyone's questions about it as I think it is important that people, particularly gays, are aware of the true nature and extent of the problem. Alongside the David Paisley situation, this week has also seen Liz Truss (as Minister for Women and Equaliies) push back against Stonewall because of their unwavering support for trans rights. This is absolutely unprecedented stuff for modern times.