I never understood why the T was included with the LGB as thought it isn't about sexuality but thought I just didn't understsnd the issues fully.
I think the main reason has been political - that you have had a number of oppressed minorities, to do with sex/gender, and hence they have been collated in terms of LGBT. In fact, there are plenty of other letters that have been added - for example, I for intersex, Q for queer, A for asexual, and there are others. There is also a historical aspect, you hear people say that at the time of the Stonewall riots (US), trans people were on the barricades with gays and others, but I've no idea if this is accurate, or kind of romantic.
But there are campaigns from time to time to split the movement, since while some categories are about sexuality, T is not. And there are undoubtedly transphobic elements, in fact, there was a fierce rumpus among feminists about this. Try googling terfs, transexclusionary radical feminists.
I find the whole movement very interesting, but it's unclear whether wholesale shifts are going on about personal identity, or whether they are relatively minor. So you hear much talk of non-binary, or gender fluidity, which can be applied to many people, I suppose, and is not trans.
I suppose right wing people and evangelicals tend to see all this as the fall into chaos and moral turpitude, whereas people on the left tend to see it as the opening up of previously suppressed areas of life.