Its basically simple diffusion. Air is going to go from high to low pressure. This can be applied to Boyle's law where it basically says that pressure is indirectly proportional to volume (a derivation of PV=nRT)...so when the diaphragm contracts (moves downwards) and your external intercostals move the rib cage up...then the volume essentially increases. By convention, because volume increases, the pressure decreases....and due to this decreased pressure you get your pressure differential! (hence air goes from high to low pressure, which in this case corresponds to inhalation). Expiration is basically the reverse of this process, but instead the volume decreases (diaphragm relaxes and rib cage moves down), which increases pressure...Hence air is passively moved from high pressure to lower pressure (exhaled).
