Motor vehicle approaches cyclist from behind. Frequent scanning of the mirror, through simple lightning-fast minor glancing movements with the eyes (rather than the alternative constant rearward turns of the head which distract cyclist from what's happening in front, much more than using a mirror, and can affect steering) alerts cyclist when it would be an opportune moment to turn around and glare at the approaching driver........and to adopt primary or execute a strategic wobble if appropriate.
Driver, who may have been contemplating a dangerous or inconsiderate move near the cyclist, observes the cyclist turning round and glaring at them, or moving into primary, and possibly strategically wobbling, becomes uncertain what the cyclist might do next, and therefore changes mind about pulling the dangerous/inconsiderate manoeuvre they were contemplating.
Admittedly not foolproof, but it will easily work often enough, reducing the number of dangerous manoeuvres, to make installing a mirror worthwhile in many cycling scenarios, particularly commuting. And of course, just because something may not work 100% of the time is not a sensible justification for dispensing entirely with that piece of equipment from the self-preservation arsenal (which some people use as a reason......not that I'm bothered whether other cyclists do or don't have mirrors).
With unimpaired hearing one can frequently tell from engine or braking sounds, and maybe stereo sound-processing of the motor vehicle's position, when a driver has aborted a risky manoeuvre near one, as well as from the visual evidence of course.....and the timing of these will indicate whether it was the strategically timed and mirror-glancing-facilitated rearward head-turn or adoption of primary that most likely induced the abort. It doesn't take long using mirrors in this way to realise that such use does indeed influence plenty of drivers' behaviour around one when cycling.
I suppose one could achieve the same by constantly turning round to look over the shoulder, plus the rest of the subsequent actions, but with a mirror it is far easier to maintain full forward visibility and awareness while assessing the rear, to reduce/avoid unwanted rear-head-turn-induced inadvertent weaves....and to conduct stealth scanning of what's behind one should that suit one's purposes too (which it can).