What causes the air pressure inside an inflated balloon?

The elastic contraction of the rubber in the balloon's membrane causes the pressure in the air inside an inflated balloon.

When you inflate a balloon, you have to expand the latex of the balloon, which stretches when filled with air from a pressure of 760 mm Hg to as high as 840 (about 10% higher than standard atmospheric pressure). If you inflate a non-elastic mylar balloon, it takes no effort: the air inside is at the same pressure as the air outside.