The TRA movement seems fairly sudden to the general public, but I think it's emerged on the wave of several different strands.
Obviously the foundation is post-modernism, which has influenced a lot of intellectual debate and thinking over the years.
Many in sociology are wedded to the idea of social construction and seem to dislike biological sciences or explanations.
Then you had the rise of Queer Theory and ideas like it is the 'performance' of femininity or masculinity that makes one a woman or man.
And any societal norms are meant to be 'queered' or transgressed.
Courses like gender studies seem to push a lot of queer theory from what I've read.
It's all been brewing since the 90s. With the rise of social media like Tumblr, a lot of these ideas and concepts coalesce into quasi belief systems that are quickly adopted by people because they have a mystical allure.
The decline of organised religion doesn't mean people these days actually engage in more critical thinking. They just transfer their need for a belief system onto another medium. There's growth in neo pagan beliefs, crystal healing, food fads, eastern exotic and mystical stuff like tantra, reiki etc and a growing skepticism of science which leads to anti vaxxers, flat earthers and so on.
Belief in human souls and the afterlife seems to have morphed into belief in a gender identity soul and the transformation into your dream body via hormones and surgery.
People want to find meaning and an explanation for their fears, discomforts and frustrations and are tempted by offers of cures and redemptions. Ideologies that offer simple explanations on the surface ( even through wracked by logical inconsistencies and incoherent woo) are very easily absorbed.
People are also sensitive to being perceived as racist or homophobic, so many folk reflexively say yes to anything related to transgenderism even if they don't know much about it or understand what it's about. It's much easier to take the least path of resistance, especially if you are constantly told that doing so will mean that you are 'on the right side of history'.
As we've seen, this means government officials and public servants will quickly adopt or change policies without giving it any thought due to the worry of being labelled transphobic.
A salesman that's trying to push you into buying something you aren't sure of.... they tell you today is the last day of the offer, it's a great deal and if you don't buy it, you're going to lose out and regret it. The tactic is to exert pressure into making a decision by implying time is running out.
The TRAs use a similar method by invoking stats that Trans people are dying left, right and centre and if you don't push through a policy or change NOW, you will regret it.