Reducing understeer into a corner
The theory here is that you can feather the brakes into a corner (trail braking), while transitioning to progressive throttle at the apex. This can lead to a very smooth transition between braking and acceleration and is less likely to unsettle the car through unwanted weight transfer. It also keeps the weight at the front of the vehicle for as long as possible, thus providing more grip, a better turn in and reduces the possibility of understeer.
Reducing drive loss through spinning wheels
This technique is particularly useful for a front wheel drive car without a limited slip differential. On the exit of the corner, it is a common symptom for one the unladen front wheel to spin while applying throttle. This spinning wheel is preventing all of the power from transferring to the road and thus slowing the exit speed. By feathering the brake with the left foot, this can prevent or reduce this wheelspin, and get a better exit.
Removing pedal transfer time
Using the left foot to brake removes the pedal transition time from brake to accelerator and vice versa. This can shave fractions of a second off a lap time when done well, but cannot be used when it is necessary to change down a gear (unless you decide not to use the clutch!)
Left foot braking is an advanced technique, and should only be attempted after lots of practice. When learning to left foot brake, you'll initially press the pedal far too hard as you'll be used to the action of pressing a clutch all the way to the floor. It takes time to re-programme the 'muscle memory' of your foot and leg, and a bit of empty tarmac is highly recommended.