Inner or subvocal speech employs what has been called the articulatory loop of working memory. Unlike a radio or TV, however, our brains apparently have only one "channel" or loop that produces speech. This channel is employed both for out-loud speech and inner speech.
Thus, a mantra, or phrase repeated over and over again, or mental noting, in which the phrase changes as you change what you note yourself doing, would tend to monopolize this speech loop. Other verbal thoughts are crowded out. This anchors you in the present moment, because it's harder to think about the past or future when words are not available.
The practice of silently noting one's smallest movements seems to go back to the historical Buddha's sayings, as collected in the Foundations of Mindfulness. We do not consider the Buddha or anyone else to be authoritative, but this particular idea seems to have a physiological basis.