Supressors are used mostly for safety purposes!
They reduce (note, not silence) the sound of each shot, greatly reducing the possible damage to the operator's hearing. In close quarter scenarios, this also benefits those around the operator. Have you ever fired an Ar-15 with no ear protection? A couple rounds and you're ears are ringing good. Imagine if you were defending your home from an intruder and did not have a supressor or ear protection; you and your family could be temperoarily deaf.
Supressors also reduce the amount of felt recoil, eliminating dangerous muzzle hop/climb, ensuring more control over the weapon and greater accuracy.
Supressors make shooting more enjoyable and less offensive to others at the range.
Supressors will ensure that no noise ordinances are violated if you're shooting on private property.
Supressors don't alert one to your location as easily as an unsupressed firearm.
Supressors are wonderful items.