These ideas have their roots in the academy and are the results of deconstruction, a literary theory, being applied to biological realities like gender. The reason these activists push this "evil ideology" as
@Erwin_Pommel put it is because they see it as the only alternative to the traditionalist view of gender, which they see as oppressive. I suggest everyone read
this blog post explaining the philosophy behind controversy. I'll summarize what it says below.
The activists in question believe what liberal philosopher Daniel A. Kaufman calls "identificationism." According to this worldview, a person is whatever he takes himself to be. So if I were to "identify" as a woman, then I am a woman. Kaufman then goes on to show how the entire exercise undermines a lot of modern liberal projects such as gay rights.
Similarly, progressive philosopher Kathleen Stock is worried about the harm done to women by this identificationist extreme.
Now why would these activists, who fashion themselves to the premiere progressives, do such a thing? The answer, Feser claims, lies in the difficulty of finding a liberal middle ground between the hyper-progressive identificationist position and the reactionary natural law position. Moderate liberals want to be able to say "things like sex, race, etc. exist as material realities that are relevant, but they do not have any legitimate moral or political valence in a modern, democratic society." They want to have to embrace reality but reject drawing conclusions from reality that would undermine the modern liberal project.
The problem lies in how difficult it is to characterize biological features except in functional terms. How can one describe an eye without referring to its function of seeing, or the heart without making reference to its pumping of blood? This is as true with sexual features as any others. Now, the existence of blind people doesn't undermine the claim that the function of eyes is to allow us to see. A blind person's eyes still have seeing as its function; it's just that their eyes are dysfunctional. If one were to apply this to sexuality, and you'd come to the conclusion that sex has a heterosexual function and that homosexuality is a disorder not unsimilar to blindness. And that's hardly politically correct.
Feser brings out the logical conclusion of this.
The problem for liberals is how they uphold the modern liberal project. There is nothing within the old natural law position or in the identificationist position with which he can do so. And until moderate liberals do come up with such a position, the only options are revolution or reaction; that is, either believing in what
@GoldRanger would call "theocratic" or "authoritarian" views (which are oh-so-spooky, don't you know?) or taking away kids from their parents to have them castrated in the name of "LGBT rights."