I attended a number of diversity learning sessions at my former company that addressed transsexual topics. Most were given by MTF trans women. One was a former body builder man that transitioned to a really beautiful woman. For us that do not have gender dysphoria, it is difficult to understand at first. But it should not be difficult to understand that these people are different from most in our society, and that they just want to look on the outside like they feel on the inside. Being different from most others in society and the discrimination that can come from that is a common issue for both trans and gay people.
Gender identity and sexual orientation are indeed different. Both are complex. But When you think about it, in the past, some of the people who were designated queer/homosexual may have really been trans people. Men who wanted to dress as women or act in a feminine way, may have been really trans and not gay. Or maybe not. Maybe they were gay men, but just felt they needed to appear as women because they were attracted to other men and society forced them to chose to be feminine acting. Either way, there is some historic affinity between the groups. Also, because the number of real trans people is so small, even compared to the small number of gay people in society, it would be difficult for them to mount any political clout, unless they were aligned with others such as the gay movement. So we are different, but we are together in the movement (what could be more "e pluribus Unum"?)
Sometimes seeing a trans person can be off-putting because their appearance may be different than our norms. But we have learned in society now to accept those who are handicapped, have lost limbs, who are in wheelchairs, who have tattoos, have nose rings and piercings, etc. No reason we can not accept trans people. And many trans people are able to change their appearances to be more conforming to our ideas of a particular gender anyway, and look really good.
Regarding Caitlyn Jenner, I think for a 65 year old, she looks really good. Being wealthy has helped, I am sure. Just as all movements have their icons that promote acceptance just by outing themselves as being part of a particular group, Caitlyn Jenner can be that person for trans acceptance. And assuming she maintains her sexual orientation, she is now a lesbian. So let's accept her as gay and as a trans person.
It's time we as gay people educate ourselves regarding our trans sisters and brothers and accept them in the LGBT movement and as fellow human beings!