What does the symbol that looks like a monster movement symbol on gate tokens mean? Or what is it used for?

When the gate is closed (or sealed) all monsters with the same movement symbol that are in arkham, the sky and the outskirts get returned to the monster cup.

So if you've got a lot of monsters out closing a gate can get rid of a few (and the symbols on the monsters might help you decide which gate to enter).

e.g. close a gate with the square symbol on it and all monsters with the square on them (and in arkham/sky/outskirts) return to the cup.

This doesn't affect any monster trophies you have - you keep them regardless of sign.

And I think you're just using the base Arkham Horror game, so this probably won't worry you, but if you're playing with expansions/custom content some spawn/special monsters get returned to places other than the monster cup according to rules on AOs/heralds/expansion etc.

Only just spotted that you'd posted two threads on this… thought it odd that nobody had answered.