The diaphragm works to expand and contract the volume the lungs occupy, thus causing respiration via negative pressure.
To demonstrate it's action, imagine a cylinder with a balloon at the bottom. If you pull on the balloon, the cylinder holds more air, and if you push in, it holds less.
Without a diaphragm, you would only be able to do costal breathing, a very inefficient method and one which is much more work. Birds and reptiles can do this, but humans need more oxygen for their bigger brain.