I'll try to unpack this as briefly as I can. Please excuse me if I go a little long.
Starting with the illustration in the OP: "trans-woman". The concept of transism rests upon the idea that 'man-ness' and 'woman-ness' is in a person's head (either as a social construct or due to brain development, I don't know what the official doctrine is this week).
Ten years ago, let alone fifty years ago, that was not an accepted idea. The words 'man' and 'woman' had existing meanings (based on obvious biology) & those meanings had analogues in all languages throughout human history. They are universal elements of human self-concept.
The left decided that "man/woman" ACKSHULY meant something else. Transism is when a person's ACKSHUL male/female identity is different from the one recognized by all of human history up to this point.
This change in the concepts behind the words "man/woman" has been pushed by academia, the media, corporate culture, and now the legal system. These words effectively have different meanings now. The original concepts no longer have words attached to them so they cannot be effectively communicated -- they are being destroyed via mandatory replacement and artificial barriers to use.
And yeah, I'm still allowed to call myself a "man." Because the language has been changed though that doesn't mean the same thing -- I can't actually say what I mean and am forced to use concepts I never agreed to couched in language stolen from me. If I say "I don't agree with this paradigm you are forcing on me" they'll say "then you're fired haha good luck supporting your family you hateful transphobic bigot."
The same basic thing has been done to a number of other important words. "Hate" is a good example. It used to mean an emotion or actions that come from it, now it means non-submission to the one true ideology of leftism.
All those concepts are no longer attached to words & thus are difficult to communicate. The new, approved concepts have been stuffed into existing words so it's hard for people to see what has changed.
That went long. I hope I was able to communicate effectively.
I hope you have a nice evening.