Picture a car rounding a corner; the inside and outside wheels have to carve differently sized circles.  Because of this, they must travel at different speeds or independent of eachother.  The non-drive wheels are no problem, but the drive wheels are connected to eachother via halfshafts, ring gear and driveshaft.  This is where a normal OPEN differential comes into play.  It allows the drive wheels to spin at different speeds when rounding a corner (the tires and drivetrain would bind and not corner as well without it).  The problem with totally open differentials is that you cannot control how much and when they slip.  Trying to accelerate on ice or gravel will usually result in only 1 tire spinning, producing plenty of smoke, but no go.

The limited slip differential does much the same job as the open, except (and nearly all OEM LSD's are like this...) there is a viscous fluid inside the differential that heats up and becomes VERY sticky/gooey when the two sides (either the two wheels or the two differentials - like on the Mitsubishi GTO and Eclipse AWD systems) spin at greatly different speeds (say, when one tire begins to spin wildly on ice or loose dirt...).  The two sides are then locked together in order to prevent all the engine torque from going to the slipping wheel.  It's basically a traction control device.  High performance sports cars use them to put power down at a standing start and when powering out of corners.  Trucks and minivans use it to keep the vehicle from spinning in adverse conditions (snow, ice, gravel, freak rain storms, etc.).  Most AWD (as opposed to 4WD) systems emply some manner of viscous limited slip in order to transfer power to the other set of wheels when needed, but avoid it to avoid drivetrain wear and improve fuel economy.  This is what Subaru means by "sending power from the wheels that slip to the wheels that grip...".   Makes it sound like there's a big decision making process when you hit adverse
conditions, but not really.