The idea is that you can feel whats happening with the front tyres thru the steering wheel, and thus you drive to the limits of those tyres.
With amazing front tyres, and ****ty rears, you can hoof it into a corner, plenty of grip and feedback thru the steering wheel, then the back end lets go, without warning, and your pointing the other way. I've been there, done it, got the tshirt (or brown trousers...), as has a mate of mine as i'll detail below.
With the stickier tyres on the rear, you know that you should never exceed their grip levels, as the front will let you know its struggling first and you'll be aware of the cars limits, and can drive accordingly.
An interesting caveat is that tread depth isnt everything. A few years ago my friend bought a corsa with bald tyrse, and picked up a cheap set of 14" steels with "good" tyres off ebay for it. We applied the above logic, and given two of the tyres looked fairly new with 6-7mm of tread on, we fitted those to the rear, and the other two (2-3mm of tread) on the front. He drove it home, found a wet roundabout and promptly spun it and had someone plough into the front of him. We were puzzled, he couldnt understand why, he wasnt driving fast and the steering hadnt felt bad/slippy etc. After some pondering thinking it might have been diesel or similar, we realised the rear tyres were bargain basement linglongs or whatever, and the fronts were michelin or pirelli or some premium brand. As a test he swapped the linglongs onto the front, and was horrified as to how little grip they had, now that he could actually feel what they were doing thru the steering wheel. With them on the back, he'd been completely unaware until he crashed.