The crisper provides a somewhat enclosed environment, which prevents moisture from escaping as rapidly. Vegetables keep best at a certain humidity, higher than that typically found in the rest of the fridge, but not so high that condensation starts accumulating on them. Vegetables kept in too-dry air in the rest of the fridge will tend to dry out and shrivel up faster; those kept in the crisper will retain their water and texture better, keeping them crisp.
Leafy vegetables are also much more prone to drying out, since they have much more surface area, while hardier vegetables with a decent skin on them (like bell peppers) don't dry out nearly as quickly. Fruits benefit somewhat from this as well, but don't generally need as high a humidity as vegetables.
Some crisper drawers have little sliders on them which vary the size of the opening to the rest of the fridge, letting you vary the amount of circulation and therefore the humidity; you can adjust this to suit what you tend to store in the drawer. If you have two crisper drawers, both adjustable, then it might be a good idea to put fruit and hardier vegetables in one, and more vulnerable vegetables in the other.