Doing a quick Google search reveals that this question has been covered quite a bit, but I found that the explanations not to be very clear or completely wrong. I thought I would try to answer this question as if I were giving it to someone with only limited automotive knowledge…
Although the concept of double clutching is easy to understand, the reason ‘why’ its needed might not be. Here’s the definition in a nutshell – Double-clutching is a manual technique used to match the rotational speed of the gears in a non-synchromesh transmission to the rotational speed of the engine.
Here’s what actually happens: The first push of the clutch is to take the transmission out of gear and then to move the shifter so that the transmission is in neutral. The clutch pedal is then released while the transmission is in neutral so that the engine is only turning the transmissions input shaft (layshaft). Now, the driver will rev the engine so that the layshaft speed is matching the output shaft speed. The output shaft speed will be determined by how fast the car is going, so the driver will have to select a gear that will keep the engine revs in a acceptable range – this process is called “Rev Matching”. The second push of the clutch is used to enter the next gear. If the driver does not have the revs matched correctly, a grinding sound will occur because of the difference in speed between the shafts which determines the rotational speed of the gears that are on them. If the shifter does move into gear, usually accompanied with a ‘clunk’ because its very hard to get the revs match perfectly, the clutch is finally released connecting the engine to the rest of the drive line.
And there you have it.
It should be noted that no modern car needs to be double-clutched. All modern manual transmissions have a device called a synchromesh which does the job double-clutching was intended to do. So, this technique is pretty much reserved for the history books except for a very few special cases.