From LEGO Batman Wiki
The grapple gun can be used to get onto high ledges using the grapple handle. In the console versions, it can be used by Batman, Batgirl, Nightwing, and Robin. Yet in the portable versions, it can't be used by Robin or Nightwing but can be used by several others like The Joker or the Riddler. It should also be noted that no one actually HAS the grappling gun as the hook simply shoots out of their hand.
The grapple gun functions exactly the same as it does in LEGO Batman: The Videogame. Yet this time, it is avalible for the entire Bat Family (besides Alfred) on each version.