It's actually the opposite of "seperate people further into groups"
They're *supposed* to be kind of umbrella terms that wrap all the smaller groups, such as "transgender, transexual, intersex, agendered, non-binary" and any other term that fits into one that can used as short-hand without excluding anyone who could conceivably fit. The more "unusual" terms, like "agendered" or "non-binary" are attempts to get the language to describe things as closely to what they are as possible. Same way that the scientific name of all the different wasps, while the umbrella term is just "wasp", if you're speaking of a specific type of wasp (or person) you use the term they tell you or that you has been ascribed to the species, otherwise you just call it a "wasp" as you *don't know* what it's actually called. It'd be no more different than calling a parasitic wasp a yellow jacket, it's just wrong by point of fact, and just because you don't know what term they prefer doesn't mean you can apply the "common term" (transgender, or "yellow jacket" wasp as they're usually the most common wasp) to them when there's a perfectly good umbrella term available to use.
And, as in your example, if you don't actually know they even are trans* at all, that's a different beast. In that case, yes, as long as you make good effort to remember, and as long as you recognize you were wrong and are now corrected with no malice on either party, then there is no real cause to yell at you. But to say "we don't need all these labels, just relax and go with the flow" is erasing what they feel, and to them and many groups in their position, feels like you're saying "I don't care what you are, just be happy with what I call you because I'm too lazy to remember another word". That may not be what you're saying, but then, you didn't mean to call someone a he who was a she, did you? But you are willing to fix that mistaken perception, but not this one?
And as "insulted to be called a member of the trans* community" (the term you would look for is cisgender, but that's generally only used as an external label from within the trans* community; there are no communities that call themselves cisgender, for the same reason there are no "straight pride" parades: every other aspect of life is cis/straight affirming, there's no need for communities catered to them because they're catered to by default), that's your prerogative, but for a lot of LGBTQ people, these communities were the first to say "No, you're not a freak, you're just different, and that's okay." There's a lot to be said for simple affirmation, and as a result, they identify strongly with the communities. And even still, just because you may not feel a member of any group, that doesn't mean others are wrong for feeling like a member.