Breath is a physiological process which can be voluntary or involuntary. When you don't try to control it, it proceeds at an involuntarily regulated pace. That's similar to blinking. When your eyes are open, they automatically blink once in a while. Breath is quite regular in its frequency, unlike blinking. Your heartbeat is also involuntary and regularly paced, but you usually cannot feel it (unless you have palpitations).
So, breath, is the only physiological process that I know of, which fulfills all these criteria: it proceeds involuntarily at a relatively regular pace and it can be felt through it's effects on the nostril, upper lip, movement of abdomen or movement of rib cage. It's frequency is also not too slow and not too fast, for observation of its effects.
Now how does that relate to Buddhism?
Well, the regularly paced change of effects on the nostril, upper lip, movement of abdomen or movement of rib cage that can be felt by a person, are sensations (vedanā). Not the breath itself, but the effect of the physiological process of involuntary breathing, causes detectable and observable sensations.
And how is this useful in Buddhism?
There is nothing mystical about the mind-breath connection in Buddhism, unlike prāṇāyāma in Hindu Hatha Yoga.
In meditation, the observable sensation (vedanā) caused by breathing is physically rooted and is something that the mind doesn't control, which the mind can use as an object to track, in order to cultivate concentration, instead of becoming bored and distracted.
If the meditator tries to observe something that his mind created, he might get carried away in imagination.
If the meditator tries to track an unchanging sensation like the sensation of his palm, he might get bored and fall asleep.
However, breathing is not imagined by the mind, and changes at a regular pace, so it makes a good meditation object for cultivation of concentration. Of course, the meditator could still get distracted or bored or fall asleep, but at least when he remembers that he had drifted off, he can come back to the observable sensation caused by breathing.