Air, naturally, is the only thing that can fill the space, so it moves into the lungs, filling them. Other parts of the body also assist in breathing. If you tighten and pull in your stomach, you push more out of your lungs because you put pressure on the underside of the diaphragm-piston. If you let your stomach pooch out, it gives the diaphragm more room to move down into the area of the soft organs, and more air can be taken in. But the muscles between the ribs all up and down the chest, especially those between the more flexible lower ribs, can also work so as to make the rib cage bigger.
Air comes in, again because there is more space to fill.
Even the muscles of the neck, shoulders and upper back assist in the breathing process. The more air the body needs, the more of the various breathing muscles go into action to create more space for breathing. The volume of the lungs is quite phenomenal when breathing is correct. The diaphragm can move up and down as much as six inches. In the process it gently massages the lower organs and aids in the blood's return to the heart.