So, LGBT+ is not exactly a rainbow of intersecting happiness of mutual affirmation. Pretty much every group in the argument has factions that absolutely hate the others’ guts.
Early on in the movement, (that is to say, immediately post-Stonewall) Lesbians and male Gays often didn’t get along, somewhat due to '70s feminism being what it was (for good reasons), and somewhat due to gay males often being patronizing, paternal, and often acting like they can speak on behalf of the whole community (a problem that still exists today to varying degrees).
Lesbians and Gays often view bi people as a liability, because their existence threatens the “born that way” narrative that LG people carefully crafted to try and get their rights recognized in the courts, by analogy to other “immutable characteristics” such as race or sex.
Trans* is the oddest one out, but is in there for largely historical lobbying reasons. In fact, two of the largest figures in starting the Stonewall riots were self-described Drag Queens or Transvestites (though they may be more accurately described as what we call “transsexual” or “transgender” after a few decades of language shift; trans identification, language, and culture has shifted a lot over 20 years). I’d also be remiss if I didn’t mention that a some straight trans people resent gay people and are very heteronormative (this is more the case in some other cultures, but even here we have Caitlyn Jenner). There’s also the splinter between binary trans people and nonbinary ones, with a few nonbinary trans people basically hating the concept of gender and binary trans people, and some binary trans people invalidating nonbinary identities or doing the “it’s too hard to get YOUR rights, let’s focus on OURS first” bullshit.
That said, trans people have also faced the largest amount of alienation. As early as 1970 people were already trying to drop trans people from the acronym because they viewed trans rights as a liability towards getting LG(B) rights due to it being “a bridge too far”. There’s also a very complicated history with lesbians and trans people, especially trans lesbians. The Michigan Womyn’s Music Festival being a particular microcosm of this, and modern TERFs in general consisting mainly of this old guard or people influenced by that rhetoric.
This is the OG LGBT, and it’s largely in tact since Stonewall (or arguably before, since the Stonewall Inn itself whose raid kicked off the riots catered to the whole acronym).
The exploding acronym is largely a consequence of two things from the 1990s: the pioneering of intersectionality, borrowed from race studies, and queer theory.
Intersectionality is the exploration of how different parts of someone’s identity may affect them and their relationship with privilege in different ways. It was originally a way to explore things like black feminism, or gay black women, but quickly got adopted into a broader context. This contributes to the acronym adopting more and more identities into it.
Queer theory is a whole… thing that I don’t think I can properly do justice because it’s an absurdly complicated field and nobody agrees on what exactly it even is. But it’s responsible for pioneering the inclusion of a lot of the “+” part of LGBT+, and why “queer” has been largely reclaimed as a catchall term.
As for specific ones, things like intersex came in by analogy to, and sometimes confluence with, trans*, and asexuality and aromantics were clearly by analogy to LGB. The history of any individual one is rich on its own, but not all of them were particularly uncontroversial from either end. For instance, a lot of intersex people resent being under the LGBT+ umbrella.
Younger queer people are much closer to the “big happy rainbow” cis straight people envision. It’s not without bitterness, but yes, you’ll find plenty of lesbians, or asexual people, or whatever, willing to champion trans or gay male rights or whatever. Some lesbians are still TERFs, a lot of cis white gay men are still paternalistic assholes, and so on.
This all really boils down to “this is a very, very complex topic that is difficult to address in a forum post”. You really need the context of a lot of reading and/or watching of different bits of queer history.