Mindfulness of breathing is a way to include the body in meditation, and it’s also as a way to ground ourselves bodily, moment to moment, during daily activities. Anālayo, in his book – Satipaṭṭhāna: the Direct Path to Realisation  (p.125)- writes:

In ancient times, and still today, mindfulness of breathing might well be the most widely used method of body contemplation. The Buddha himself frequently engaged in mindfulness of breathing, which he called a “noble” and “divine” way of practice. According to his own statement, even his awakening took place based on mindfulness of breathing.

To obtain a sense of the importance of mindfulness of the body, we only have to consider that attention to the presence of the body is a foundational ‘step’ in the following practices:

  • awareness and investigation of the five ‘sentient processes’ (my translation of ‘khandhas’); 
  • the four establishments of mindfulness (satipaṭṭhāna); 
  • and the ‘Mindfulness of Breathing’ meditation practiced by the Buddha (ānāpānasati);
  • and, related to the first point: in general in the teachings (not only in the first foundation of mindfulness), mindfulness of the body is important in the contemplation of the earth-element, as well.

The body also figures in what the Buddha called ‘the All’; ‘that is, in the six’ channels that he said comprise the whole of our lived world (which, by the way, we moderns can expand. I write elsewhere of seven.). This is an experiential thing. The ‘bodily sense’ in this context is often translated (perhaps correctly, in a strict translation) as ‘touch,’ but this is often an inadequate term to describe the fine and intricate body-senses which are available as objects of our mindfulness. Just consider, for instance, the existence of (what we now call) proprioception; and of kinaesthesia; and also of the ‘felt sense’ (Gendlin). The importance of the body as the early or first steps in the above practices is that founding our awareness in our ‘physical’ presence gives us yogis a steady basis for the realisation of progressively subtler states of awareness; and it allows integration of these subtle, spacious states with our daily activity. So, moving ever deeper into subtle phenomena – such as the feelings, mind-states and the dynamics of suffering & liberation – does not mean abandoning the body after its initial role.
Anālayo points to one way the awareness of breathing is integrated. He points out that during the sixteen steps of Meditation on the Breathing Body [of ānāpānasati], all bodily or mental phenomena coming coming into attention:

are experienced against the background of the ever-changing rhythm of in- and out-breaths, which provides a constant reminder of impermanence. (Anālayo, p.134)

In other words, as we progressively encounter subtler phenomena in ourselves, we continue to check in with the breath and body; we transcend and include the body, embodying our realisation.