What happens when you press the clutch pedal?

(Assuming you have a hydrolic clutch system)

When you push on the clutch pedal, you create hydrolic pressure in the master and slave cylinders, which in turn pushes on the release bearing (AKA throw-out bearing). The release bearing pushes on the fingers of the pressure plate. This takes pressure off of the clutch disc, putting the car in neutral, in a sense. Now you can change gears without damaging any of the transaxle gears.

When you release the clutch, you put pressure back onto the clutch plate and pressure plate, which gives power back to the wheels if you are in 1st-5th gear, or reverse.