When it contracts, it pulls down, increasing the volume of the thoracic cavity, which increases lung
volume, which causes a decrease in pressure. Air always follows a pressure gradient (from high to low), so this causes air to rush in.
Two of the main nerves resposible for breathing are the Phrenic Nerve from the cervical spine (neck region) to the diaphragm, and the Vagus Nerve from the medulla. Brainstem participants are the medulla and pons.