For many applications a base that was made only of copper would probably be too effective at transferring heat. Heat applied to a small region would be transferred so rapidly that it wouldn't have time to diffuse across the pot's bottom. This would result in uneven cooking and possibly even local areas of scorching. Capping the base with stainless steel slows the immediate rate of heat transfer from the burner or heating element, but once this heat enters the copper core its high conductivity would spread the heat rapidly and evenly to all parts of the base. There are also some esthetic issues behind this design.
Copper is notoriously hard to clean, while stainless steel is, well, stainless.
The thin, exposed band of copper near the bottom is no doubt there to catch the eye of wandering customers.