The possible rationales would be firstly,that when all one’s attention is consumed by an activity there is no room for ruminations, anxious thoughts, forward planning, self criticism and whatever else fills the mind. Perhaps all that background aggravation actually triggers low level fight/flight response with its cascade of unhelpful consequences in the body especially when there is no where for those hormones, adrenalin and such like to be burnt off.
Regular respite from the background story of our minds therefore, would give the body a chance to restabilize itself but perhaps more importantly this would happen more quickly than if it is frequently in flooded by the fight/flight response with no respite.
The second possibility is the actually focusing attention and by default blocking out background chatter actually triggers the relaxation response which more actively engages the parasympathetic nervous system to calm the body down.
The possibilites for what practice or activity would produce the benefits could range from undertaking an ATM to being playing in a 5 aside match, climbing a rockface, practicing mindfulness, doing yoga, listening to Bach’s 5th symphony, sitting in prayer, whatever fully engages all of you.
How regular is regular? I would imagine the more regular the better but a little is better than none. Like most things its how to build it into daily life so that its one of the things that are part of your routine and therefore, are less likely to be ditched at the first life wobble. For most of us it would take a catastrophic life disturbance that to stop the habit of brushing our teeth.