When the diaphragm contracts (it's a large flat muscle), it pulls the lungs down this causing expansion which eventually lead to inhalation of air. If it's injured, then that will be affected.
A complete spinal cord injury affects everything from the injury down. This means that if the diaphragm is affected, so are the intercostal and abdominal muscles. Without the use of these muscles, a person cannot breathe as easily or cough or sneeze well