The two biggest reasons for a good stabilizer are for balance and noise/vibration dampening. The good balance is important because you hold your bow below the mid-point of the riser, resulting in it being top heavy. The added weight of the stabilizer on the bottom returns good balance and makes it easier to hold the bow straight, using less muscle control to keep it straight. That way, you don't get as tired as fast. It's not a big deal for the one shot you get hunting, but it's real important for all the practicing that we should do before we get into the field.
The vibration dampening is important for a couple of reasons. First, less noise means less chance of the deer or whatever "jumping the string." Of course, you can also use limb savers, and other types of things that are much cheaper to beat the sound. I use a stabilizer and a total of six sound dampening devices (two built into the riser, two limbsavers, and two string silencers). Second, the vibration dampening is important to reduce the hand-shock you feel when the string is released. As far as I can tell, it's the equivalent of the recoil from shooting a gun. The stabilizer helps absorb that vibration in the riser so your hand and arm don't have to. That way, the muscles in your hand and arm don't get as tired from having to react to and absorb the vibrations. Again, absorbing shock isn't a big deal for the one shot while hunting, but makes practicing a lot easier. And they always say, perfect practice makes perfect.
Just my thoughts on it. Take them for what they're worth. Hope they help.