Left-foot braking is a technique that suits a variety of situations, from circuit racing and rally driving to certain off-road manoeuvres in older, simpler vehicles.
In circuit racing and rally driving you should (with very few exceptions) be either accelerating or braking, (almost) never coasting, so most competitive drivers will left-foot brake regardless of the vehicle and driveline. It lets them overlap the use of the pedals to eke out a tiny bit more corner grip (load transfer), as well as get back to squeezing on the power just that fraction sooner.
Left-foot braking with a gearbox that wants you to use the clutch is a little more tricky, but it’s a dance that can be done. Plus, in competition vehicles not every corner needs a gearchange, and not every gearchange needs the clutch pedal to be depressed.
The technique’s usefulness also extends to off-roaders, but for a different reason.
When electronic aids are deployed this method becomes redundant (because the on-board computer will do this for you), but on old-school 4WDs the trick of left-foot braking whilst simultaneously modulating the use of the throttle is a simple form of traction-control. The idea is – once you’ve had some practice at it, and I recommend doing a course – you’ll be able to keep the vehicle moving through slippery uneven obstacles with your right foot, and contain the wheelspin from the tyres that have the least grip by using your left.
FWD race and rally cars may also use left-foot braking for the same purpose of simplistic traction control out of corners, but at much higher speeds obviously.
For those used to driving a manual in the conventional fashion, using your left foot on the brake is a tricky thing to wrap your head around at first.
The action of operating a clutch usually requires depressing it fairly quickly, and releasing it at different speeds (slowly for a standing start, and fairly swiftly for a gearchange).
The action of depressing the brake however, requires modulating the application for the grip available. Stab it and the front brakes will probably want to lock up (or activate the ABS), so even when you want to brake heavily you can’t just stand on the pedal (usually). You need to be firm, but not all the way to the floor in an instant.
Your brain will gradually rewire itself though, just as it does when you learn any new motor skill, and you’ll get better at it with practice.
Left-foot braking on the street is generally discouraged though, and fair enough too.
There’s no valid reason for it (because you should never be in that much of a hurry on public roads), and just as learning how to left-foot brake after years of using the clutch is a challenge, so too is learning how to use a clutch after getting used to using the left foot for the brake all the time in an automatic.
Moreover, an emergency stop (or any stop for that matter) in a manual requires depressing both the brake and the clutch, and if the reflex action is to press the brake with the left that can make for a compromising situation. It would likely result in a longer stopping distance because the engine is still pushing until it stalls, and when it does stall it leaves you stranded (and potentially a dangerous obstacle) for at least as long as it takes to realise what happened and get it started again.