Dogs may instinctively respond to belly rubs using ancient behaviors. Wolves and other canines display their bellies to submit to pack leaders in the wild. By accepting belly rubs, dogs trust their human "pack."
Belly rubs can help relieve itchiness in dogs. It's hard for dogs to scratch or rub their belly on the ground, so having someone do it is handy.
Dogs' bellies are fragile, therefore they may seek warmth or protection from their owners. Belly rubs can reassure.
A dog that likes belly rubs will likely request them again. This favorable relationship deepens the dog-owner attachment and promotes the behavior.