Mindfulness of breathing meditation (ānāpānasati) was originally practiced by the historical Buddha Gotama and was used by him as the basis for his own practice, after which, he proceeded to insight meditation (vipassana) through which, he achieved Enlightenment, Nibbāna/Nirvāna.
He taught it repeatedly, praising it in many of his Discourses; on how it can bring about deep concentration, calm, serenity, relaxation, bliss, and equilibrium or composure of the mind. Thus he says:
“This concentration (samādhi) through mindfulness of breathing, when developed and practised much, is both peaceful and sublime. It is an unadulterated blissful abiding, and it banishes and stills evil, unwholesome thoughts as soon as they arise.” (S.V.X.i.9, Vesālī Sutta)
Breathing is a very vital part of our existence. When one takes it as an object of meditation, it is very easy to concentrate, because it is repeated over and over again. This repetition gives the mind a fixed point to concentrate, and one can easily gain unity and focus of the mind. Thus, the mind is unified, tamed, and hindered by any other thoughts that may be distracting it.
The ordinary mind is in an unrest, scattered and unstable state and can easily experience stress, anxiety, discomfort, etc. With this meditation one can bring it to a state of inner harmony and serenity, deep relaxation, and calmness. Therefore, the purpose of this meditation is to focus the mind so that, over time, the meditator can understand that his/her scattered mind becomes slowly unified and concentrated.
Moreover, constant practice in the meditation on breathing can lead to the attainment of the ecstasies or absorptions (jhānas), where the feelings of joy, bliss and equanimity grow exceedingly to the point of climax. Besides, the high degree of concentration and equanimity the ecstasies produce can be used for insight meditation, through which, it is possible to attain Enlightenment, Nibbāna.
Bhante Nynadassana explains in great detail this meditation exercise based on the authentic instructions that the Buddha gave to his disciples 2,500 years ago.