I posted this (slightly rewritten) on another thread but fits here as well I think...
I think many liberal/progressive people hear this "poor trans people suffering in the wrong body, living a lie" narrative, and superficially it seems very close to other oppressed groups who fight against the social role/stereotype imposed on them by society so they instinctively feel it's something they as good people should support.
Tag-teaming LGB has created such strong associations with "the right side of history" that they don't consider it worthy of challenge. The assumption is that the arguments in favour of LBG acceptance and rights were fair so the arguments for whatever TRAs are demanding must be as well. They aren't looking at the details.
So it never occurs to them to ask "hang on, how does this idea that a man can born in a woman's body and vice versa fit with the general acceptance that saying men and women are born with different types of personality is regressive sexism? Where exactly is this mannyness or womanyness coming from?" Or "has anyone actually proved that trans women's social role and behaviour is close enough to female people's to justify lumping them together legally, statistically and socially?"