The purpose of the rear differential is to transfer the longitudinal rotary torque of the prop shaft 90 degrees to the wheels, while allowing the wheels to spin at different rates for when going around curves (the outer wheel needs to travel further than the inner). Here's how a differential works in general terms.
The input shaft spins the cage which tumbles the freely rotating blue "spider" gears. As they tumble, the green axles are spun. When the spider gears spin on their own axis, it allows one of the green shafts to turn at a different rate than the other.
In situations where the tires have different traction, a regular "open" differential as described above will direct all the torque to the tire with the least traction. This is a bad thing when going around curves as weight will transfer to the outer wheel and the inner wheel will lose traction. So just when you want more torque going to the outer wheel that has traction, it won't go there, and the torque will just spin the inner wheel uselessly. Likewise for when starting on snow and ice and one tire has more traction than the other.
For this reason, various "limited slip differentials" have been invented over the years. They use clutch plates to lock one of the axles to the cage thus preventing the blue spider gears form rotating about their own axis. Until recently, these have all been of mechanical design. The innovation in the M5 is to have these clutch plates operated by an electrical motor under computer control.
A cutaway of the unit is shown below.
The input shaft enters from the far side. The rear wheel axles are connected to the sides. The protruding device on the right is the computer-controlled motor which tightens and loosens the clutches. Here is another view.
We can see the electric motor which tightens up the clutches.
In this image, we see the inner workings at the front. The large bearings and worm gear that turn the cage are seen clearly.
The control unit (1) is under the trunk of the car near the battery.
It communicates with the central gateway module using the new FlexRay car computer bus standard, indirectly getting information from the stability computer (wheel speed, target transverse torque distribution, stabilization status, braking value), the motor electronics (accelerator pedal angle, wheel drive torque, "engine running" signal), and the integrated chassis management computer (wheel circumferences, lateral acceleration, yaw speed, vehicle speed, roadway inclination, steering angle).
Every 1000 km a quick calibration is run to correlate locking torque with motor current, and also assess clutch wear. The diff contains its own fluid, cooled by a heat exchanger, which are the veins underneath the unit in the under-vehicle airflow.
The operation of the limited slip rear differential is very noticeable when accelerating hard out of corners, especially in situations where the traction is limited. In my previous car, the E60 545i, the traction control would cut in in these situations, applying brakes and cutting engine power. Without traction control, you would need to accelerate out of the curve slowly or risk losing the rear end. In the F10 M5 by contrast, the car feels amazingly composed at speeds which would have easily spun out the other car.