Does this mean if a monster moves on a space with a character on it during the mythos phase and the character fights it on his next turn that he doesn't move during that turn?
When an investigator starts movement in a space with a monster, he has to decide if he wishes to leave that space or stay and fight.
If he wishes to leave, he must make an evade check for every monster in the same space. If you evade every monster, you can leave the space. If you fail, you receive the combat damage from the monster and stay and fight the monster.
But if you fight the monster (either as a direct choice or as a result of failing the evade check), your movement is over because combat has begun.