What nerve carries impulses that stimulate the muscle fibers of the diaphragm to contract?

The answer is phrenic nerve. Originates in the neck (C3-C5) and passes down between the lung and heart to reach the diaphragm. There are two phrenic nerves, a left, and a right one hence called a bilateral nerve. It also contains motor, sensory, and sympathetic nerve fibers hence called a mixed nerve. It is the only source of motor innervation to the diaphragm, hence critical for breathing.