This means staunchly combating the reactionary subsections of the trans population overtly influenced by gender roles in their decision to transition as this only serves to reinforce gender roles and provide a smaller space in which womanhood or manhood can reside.

This is a part I have problems with. I believe it's saying that people who consider desire to fulfill traditional gender roles in their decision to transition are reactionary. Like: I want to be a housewife so I need to be a woman.

I'm not sure that anyone actually embarks on transitioning on such grounds and even if it affects someone's decision then that's up to them and it's totally valid.

Radical feminists rightly criticise this to some degree but draw the wrong conclusions, and liberal feminists fail to engage with this at all.

Rads believe transitioning is either mostly sexual or about appropriating gender roles. They reject the appropriation as male privilege. Liberals don't really think about gender roles beyond providing cis women access to male gender roles as empowerment.

We must be very careful in our own criticisms, however, as some people take any trans woman’s display of femininity as automatically illegitimate, as a sign that they’re “faking” or “appropriating” womanhood, or that they’re some sort of drag queen.

This reads pretty straightforward.

This means wholly rejecting the biological-essentialist “brain sex” framework that liberal feminists cosy up to, but radical feminists rightly oppose.

"Brain sex" refers to the liberal conception of sex as being biological and innate. Essentially that old cliché "a woman trapped in a man's body", which was a useful analogy when trans people were struggling to be distinguished from cis queers, but is not entirely accurate.

The prevailing notion – at least in liberal feminism – that dysphoria causes transness is wholly wrong. An equally wrong idea, popular for radical feminists, is the view that transness causes dysphoria.

This goes back to what I just mentioned about "brain sex". In the liberal conception, gender is innate to the brain, conflict with the body causes dysphoria, the way to fix dysphoria is by transition. In the radfem conception, sexuality or appropriation create the desire to be a woman, the desire causes bodily dysphoria, the dysphoria causes the desire to transition.

There’s no one-way causal relationship; instead, they co-emerge. They’re in a feedback loop, forever remaking, reforging, and reshaping each other. Utterly inseparable, yet distinct at the same time. The spiral metaphor applies very aptly here as well.

This is also pretty straightforward.