Meditation on breathing was used by the historical Buddha to develop the concentration of his mind to a great degree. This technique helps the meditator develop their mental concentration and acquire spiritual serenity.
Mental concentration and serenity were used by the Buddha as the basis for insight meditation (Vipassana), through which he reached his final Enlightenment, the Nibbana.
Continuously practicing meditation on breathing can lead the meditator to meditative ecstasy (Jhāna). Ecstasy is a state of extreme joy, happiness and bliss, but it is always subject to the law of rise and fall; therefore, it cannot lead us to Enlightenment by itself, but it can help pave the way toward it.
Bhante Nynadassana explains in great detail this meditative exercise, based on the authentic instructions that the Buddha gave his pupils 2,500 years ago.