How does the diaphragm work to change the air pressure in the lungs and, as a result, cause breathing?

When the diaphragm contracts and moves lower, the chest cavity enlarges, reducing the pressure inside the lungs. To equalize the pressure, air enters the lungs. When the diaphragm relaxes and moves back up, the elasticity of the lungs and chest wall pushes air out of the lungs. hope this helps