They said that the people on the blacklist will remain there, you just won't know their names. You can take a screenshot of the names so that you remember, though. Press Print Screen, then open Paint in the windows Search bar, then CTRL+V, then CTRL+S.

To register someone on the blacklist, you can:
  • Visit their world, Player Search them, then right-click and Add to Blacklist.
  • Right-click them in the chat window.
  • Right-click them in the battle activity window if you saw them fight things.
  • Right-click them in the Contact window (people you recently partied with) and Add to Blacklist.
You can also go to the lodestone and search for them there and block them - although it's unclear, it suggests it may also block them in the game and vice versa, but I haven't tried myself.