It slows you down (braking), saves your clutch from burning out completely (blipping throttle to rev-match) and makes you look really cool when you do it
Seriously though, say you have a long straight, followed by a tight right-hander. You are doing about 100mph in 4th gear, then you brake hard, slow down to the right speed for the corner and then change down to second or first. If you do this a few times you will seriously melt your clutch.
Using heel and toe, you are on the brake the whole time, but dropping gears whilst you are braking with the aid of the throttle. This means that there is virtually no wear on the clutch, and as soon as you come off the brakes, you are in the right gear.
It took me about a year to learn the technique, and about another 6 months to put to practical use. The 172s pedals are built for it.
However - I learned heel and toe in my old 1.2 8v Clio
. Start by going along in 4th gear and change down to 3rd without braking.
Look at your revs in 4th, and guess what they would be if you were travelling at the same speed in 3rd.
Dip the clutch and change into 3rd, whilst simultaniously blipping the throttle so the revs shoot up to the revs you estimated.
Release the clutch (you can almost sidestep if you matched the revs correctly)
If you did it correctly you should not have felt a jerk (or a lunge if you revved too much)
Master this, then try it braking too
Have fun.