Air exerts pressure because its molecules are in constant motion; they impact against objects they encounter and exert a pushing force against them. The force of multiple impacts over an area is air pressure. More air molecules from denser air will increase pressure, as will faster movement from high temperature.
Air molecules move in every direction, impacting objects and other molecules in the air itself. While the direction of any particular air molecule is basically random, air molecules travel further when there are fewer air molecules in their way. This is why air tends to flow from areas of high pressure to areas of low pressure. However, gas molecules are very widely spaced from one another, and so impacts are relatively rare.
Gas molecules are well mixed in air, and there are only a few weak forces between them. Thus, the pressure they exert is often approximated as being purely a function of the temperature of the gas and the volume of the container of the gas. When the volume of a container is increased, the pressure of the gas is decreased because the molecules move further apart. This is because a gas always expands to fill any container it is placed in unless another force, such as gravity or the electrical force, keeps it from doing so.