How does air exert pressure?

Different gas molecules make up the molecules of air. The container walls and the gaseous molecules are constantly coming into contact with one another. A tiny force is applied to the wall when a molecule strikes it. The total of all these impact forces with the container wall results in the pressure that the gas is under. The pressure increases as more particles collide with the wall.
Gases therefore exert pressure because they clash with one another in the air.