Its just an additional aide.
As mark says, with quattro, the only diff that can "lock" is the centre, so if you park one side of the car on ice/grass/mud, and the other side on tarmac, and try to drive off, the diffs in the front and rear axle will send the power to the wheel thats easiest to turn. That means you dont move, as all the power goes to the side thats parked on the grass. The same thing occurs in landrovers with the centre diff locked, which is why proper mud pluggers also fit locking front and rear diffs that can be controlled from the cab.
The EDL just aims to help out, by applying the brake on the slipping wheel, The amount of torque applied to the slipping wheels brake disk should be passed thru the diff and delivered to the other wheel. You cant completely stop the wheel spinning, because a) the engine is more powerful than the brakes, and b) locking the wheel completely would impede forward motion.
Interestingly, most newer defenders implement a very similar EDL system, presumably because its more effective than nothing, and much cheaper than actually installing locking diffs at both ends.
The thing about getting a quattro sideways kinda makes sense too, in general when you accellerate weight is transfered onto the rear and off the front. If its slippy that means the front will spin up before the rear, the Torsen in the centre diff then sends more power to the rear end to try and regain traction on the front, and the increased power delivery to the rear causes the **** to step out.