The mechanism of breathing.

When volume is increased pressure is decreased.
Air rushes into areas of lower pressure.
So increasing volume in the lungs brings air in:

Breathing in (inspiration)
Diaphragm contracts: flattens- increasing volume.
Intercostal  muscles contract: pulling up and out- increasing volume.

When volume is decreased it increases pressure.
Air rushes to areas of lower pressure.
So if volume is decreased in the lungs air will rush out:

Breathing out (Expiration)
Diaphragm relaxes: becoming a dome shape and pushing up- decreasing volume.
Intercostal muscles relax: pushing the ribcage down and in- decreasing volume.