Basically (really basic), rev-matching is about matching not only the car speed to the gear you are in, but also matching the engine speed (revs) to the speed. When you are accelerating, for example, and open the clutch (step on the pedal), revs drop for a brief time while you take the time to actually shift. If you shift very quickly, you almost get no perceptible "lunge" as the clutch is let off and the next gear is selected. Rev. matching subscribes to the fact that you should "blip" the throttle just as soon after you push in on the clutch or as soon as you're just about to shift and let off the clutch.
This basically smooths out the transition from gear to gear. Some people like the "kick" when shifting; other people like to rev-match to make the drive as smooth as possible.