Besides doing some damage to Players, the main function of the Grappler is pulling you towards wherever you shoot quickly. Its ammo (20 at the beginning) cannot be found, making a Grappler without ammo useless. It can also be useful to chase a VIP or any runny enemy faster.
Use this weapon to surprise your enemies and launch a quick attack. When you are being pulled towards the spot where you shot the Grappler, you will move towards that spot in a high speed. This quickly, sudden movement may catch the enemy off-guard. When you have moved to the new spot, instantly fire your weapon at the enemy. You can basically use the Grappler with any sort of gun, making it a useful tool to take with.
Here is a good tactic. Grapple yourself straight at the enemy and switch to a shotgun (pump shotgun is recommended because it deals a lot of damage per shot). When you get close enough, blast your shotgun in the enemy's face, potentially knocking him down (or eliminating him in solo).
It is also good for escaping as you can use this weapon to pull yourself away from the enemy quickly. Just make sure you don't rush into another enemy!