Stabilizers improve accuracy, especially at longer distances. The idea is to #1 better balance the bow, as more often times than not they are top heavy because of sights, rest, etc. And #2 they dampen the effect of minute shifts in bow orientation that occur in the milliseconds after the arrow is released before it has fully left the bow. If you've ever shot at 20 yards and had arrows in a circle 2" off the x-ring, but none in the x-ring, you've experienced this phenomena.
In hunting, they add confidence in the shot and prevent arrows from flying too wildly. Things get tense in the seconds leading up to a shot, and most likely you've sat in a stand or blind for awhile without shooting the bow or warming up, and you're taking a fresh shot. Sometimes reduction of that small margin of error can save you an extended track job on a hunt.
I've hunted with and without with success.