If there is no one else in Manhattan (i.e. it's the beginning of the game), then the first player to roll and keep claws will go into Manhattan. When entering Manhattan, you always start in Lower Manhattan.
You may be forced into Manhattan if you attack a monster in Manhattan and they flee. They then enter your burough and you enter Manhattan during your Movement phase. The same applies if you eliminate a monster in Manhattan. They are removed from the game and you enter Manhattan during your Movement phase.
If there are no monsters in Manhattan (or the 5-6 space is free when there are 5-6 players), the player who begins their Movement phase next enters Manhattan. If there are no monsters in Manhattan at the end of your turn due to a card effect resolved after you moved, you are forced to enter Manhattan once your Buy Cards phase is over.