I found it hard to pick out many words from all the simultaneous talking and can only hear part of the much requoted phrase.
He evidently had trouble hearing her too as he challenges her "Did you just call me a slave?"
Requesting another passenger to move to another seat, to suit oneself, is completely unreasonable. Superficially, it's racist but we don't know if she would have asked someone of her own ethnicity to do likewise, given the same triggering actions as here (not captured on video).
I am slightly curious as to why the person with the phone began filming so far ahead of the provocative statement being made and 4 minutes eats a fair chunk of phone memory. Clearly we are missing actions which were sufficiently remarkable to inspire someone to bother starting filming.
Even if the comments were factually correct, they were tactless, tasteless and culturally insensitive. If he's from Africa, not Caribbean, then she is factually wrong.
The seemingly pre-thought-out ideas about the chip on the shoulder attitude are a generalism (always a bad start) which she appears to be attributing to his ethnicity and, if she uses that as an excuse to treat him differently from other people then I believe that to be racist.
"Walking on eggshells" with minorities is also treating them differently and thus racist.
If your attempts at humour were received badly in the past and this is why you began "being careful" what you say around then carefully consider your entire thought processes about them. Better yet, make friends with them and try to broach the fact that you facked up, in your dim and distant past.