I have also been trying to understand the thinking behind this claim. As a general principle, if I'm disagreeing with someone I would always want to make sure that the thought process behind their statements is what I think it is. (And it was realising that I was failing at this with the gender-critical feminist approach that set me off on the route to peak trans.)
It is difficult to follow with this one though because it seems that often, it's a statement of affirmation rather than logical conclusion of an argument. It's saying "I believe trans women feel how they say they feel, and I am willing to expand my own social category of 'women' to include them." Opposition to this is perceived as a) refusing to believe them, and b) unfairly refusing access to that social category. So when they hear "but trans women can't be women," they hear it in the sense of "but non-white people can't be British" or "but a woman who's been married to a man can't be a lesbian" - it's both bigotry and refusing to accept the validity of someone's feelings in favour of your external perceptions.
As for why feelings get a higher status than biology on this one I'm less clear. As best I understand it, the people who believe that trans women literally are women (rather than just, should be included in the same social category as biological women) would view it in the same way as intersex conditions. So, if someone was born with CAIS (was biologically XY but developed as female) and did not discover this until puberty, it would be cruel to say "but you can't be a girl, despite your female body and your upbringing as female and your belief that you are female - your chromosomes say no." For a trans woman, the physical body & upbringing as well as the chromosomes might be male, but she has the same belief as the intersex woman that she genuinely is a woman and therefore it would be the same level of cruelty to disagree with her. Throw in a heavy dose of beliefs that 'biology is complicated, sex is on a spectrum, nobody knows what their chromosomes are anyway'* and bingo, trans women are women?
For this to hold up it seems to me that you do have to accept gender identity as some kind of inevitable, internal state, and at that point I don't follow. I also don't follow why this same logic wouldn't apply to e.g. Rachel Dolezal as well as Bruce/Caitlyn Jenner. But as far as I can follow it, this is what underlies it.