Okay, here I am, as @thenewgreen requested!

You've gotten a lot of really great answers so far - shoutouts especially to Kafke, Quatrarius and khjuu for great answers though I don't agree with everything everyone said all the time (but that's me and my opinion).

Since you've gotten so many good answers, I think it might be good to flesh out some of the history, and provide some context from that perspective.

While there were earlier events, The "shot heard 'round the world" of the Gay rights movement in North America is the Stone Wall Riots. Basic story - Mob-Owned Gay bars were paying off police to prevent raids (running a "gay bar" was illegal at the time, and the mob was pretty much the only game in town willing to do it). Stonewall was seen by many as "the" gay bar in the city, but it mostly only catered to Gay men (with a few Lesbian clients), and was one of the few bars which catered to effeminate men, cross dressers and trans women (though AMAB people dressing entirely in women's clothing was discouraged - and AFAB women had to be wearing "at least 3 articles of women's clothing"). The alcohol was bootlegged, raids were frequent.

This raid, At 1:20 AM on Saturday, June 28, 1969, was different. Instead of the usual seizure of alcohol and taking down of some names, maybe arresting the most flamboyant, it began to look a bit like a mass arrest. A crowd grew outside, mostly of gay people, and the mood darkened. After a woman (possibly Stormé DeLarverie) was brought out in handcuffs and evidence of being beaten, it turned into a riot.

At the core of this riot were the crossdressers, effeminite gay men, and trans women. women like Silvia Rivera, Marsha P. Johnson and others, along with many Gay, Lesbian, Bisexual and queer people were at the heart of this movement. The T is in LGBT because historically, we were a big part of the history of said movement.

So that's some history into the "why" of it.

The question of whether we should be there or not is sort of a more complicated one. One one hand it ends up spreading a lot of bad ideas about trans people - that we are just extra gay men and women, or that we transition because we are so ashamed of liking the "same sex", or (especially in the case of trans lesbians) that we transition because we love the opposite gender so much we just want to be them. On the other hand, as rjw says, a lot of it has to do with intersectionality. We have a lot of legal needs in common - the right to marry, the right to live as ourselves without fear of persectution - so there are benefits to working together, and we have our similarities: we are all humans who are "fringe" cases, a small part of the population.

I hope this helps provide more context into your questions.