I'll try to answer the question of exactly how the clutch works. In essence, the Shimano clutch design uses a one-way bearing that allows the cage to freely rotate clockwise, but adds friction if rotated counterclockwise. When activated, it increases the torque needed to move the cage by around 3-7Nm.
(Picture from BikeRadar)
The clutch lever (orange) rotates a cam (oddly shaped metal piece on the right side), much like what flipping the lever on a QR skewer does. This makes the tensioning band tighten around the central clutch unit (cylinder-like thing with the octagonal metal piece on it). The cam presses the tensioning band against the tension adjusting screw (uppermost piece), which by screwing it in and out allows you to change the clutch stiffness.
I'm not 100% sure what happens inside the clutch unit, but I believe it's a roller bearing cam system. There are long, cylindrical needle bearings inside which are constrained by specially-designed slots. They can freely rotate when the unit is spun one way, but they jam and lock the unit when spun the other way. I believe that in the jammed direction, the outside of the clutch unit rotates in the tensioning band. The compressive force creates friction here.
This explanation is supported by how Shimano instructs you to lubricate the outside of the clutch unit, but to leave the inside dry. These types of roller bearing clutch don't work very well (if at all) when lubricated.
Here, you can see the axle (with the 4mm hex fitting) around which the clutch unit rotates, and then the clutch unit separate from the tensioning band (Images from NSMB):
Here, you can see the inside of the clutch unit. Note the skinny little needle bearings (MTB Direct Australia):
Note that the clutch does not add extra tension to the chain, as some people may claim. It is a passive unit which simply prevents rotation--it doesn't actively apply a spring force or anything. As for shifting feel, I do notice that shifting feels slightly heavier with the clutch on, but there's no obvious reason for why this is. Perhaps some amount of cage rotation is needed when moving the derailleur side to side?