Just a quick response to the opening question as so many have posted very good summaries and links and this thread looks like one that should be saved.
I think many feminists dont as such have a problem with "transgenderism" ie that some people may want to express not being trapped into the social conformity that says if you are male your dress and behave in certain ways, and the same for women.
But the issue for feminists is that what had seemed to be something that was part of the "gender bending" of the 70s has turned into a campaign and set of beliefs that centre on 2 things. One is that you can change sex. The other is that being a woman is a feeling.
Feminism and specifically the feminism of women's liberation is that women are oppressed on the basis of the sex they are born. And that to say that someone who is born of the sex class that oppresses women ie men, can become a woman by identifying as such is not only a denial of biological reality, but also an attempt to erase the reality of the discrimination that women face because it denies the reality of sex.
Many of us were not aware of, or did not forsee the consequences of the GRA 2004. And it might not have been much of a disrupter had it not become (or as some say was also intended to be) the foot in the door) to become an attack on women's sex based rights.
I know some people do not like this comparison, but it is noticeable that most people rejected the notion that a white person could claim to be Black because they identified as such. To accept that would be to deny the reality of not just daily racism, but the experience of growing up in a racist society and how that has impacted on your life.
Probably not for this thread but there is also the fact that the arrival of queer politics in universities in the 80s which led to the dismantling of women's studies and more, has created generation of women who have been "educated" into believing that sex and gender are the same thing. Many of these are now in positions of influence in the media etc., who now routinely report on someone's gender in a news report, rather than their sex.
In doing this women are being erased. If society is conditioned into thinking you dont exist, how can you campaign or legislate to protect your rights (one of the aims of some trans activists was to get rid of Sex as a protected characteristic in the Equality Act).
Gender is a social construct that most feminists want to dismantle. Sex is a biological reality.
Transgenderism as now expressed and echoed by the media has become an attack on women's rights because it denies the existence of women.