Basically, it's because there are
two separate "rev-matches" that need to be made. We're all familiar with rev-matching the engine flywheel with the clutch plate when releasing the clutch. But it's also necessary to match the internal gearing of the transmission in order to shove the thing into gear in the first place. Modern passenger cars almost universally have "synchronizers" in the transmission that take care of this for us, which is why double-clutching is not necessary.
This article at howstuffworks.com gives a pretty decent picture of just what's going on.
Basically, with the clutch disengaged and the transmission in neutral, the layshaft and the actual gears in the tranny will spin to a stop, as they're now connected to neither the engine nor the wheels. To physically put the car in a gear, you have to mesh a collar with "dog teeth" on it to a gear fixed to the layshaft. This collar is spinning with the driveshaft.
To match 'em up, you need to "blip" the layshaft up to speed. Since the layshaft isn't connected with the engine when the clutch is disengaged, you need to let out the clutch briefly to "hook them up," and
then give the throttle a blip.
Reading through the link I provided will hopefully make things more clear.
This whole practice is purely theoretical to me, BTW.

I've never driven a race car, or even a car without synchros so I have
zero actual experience with double-clutching.