Double declutching (not clutching) is used to match the rotational velocities of the gearbox components so that they mesh cleanly. It's important in a non-synchromesh box, you get horrible jerky changes without and the box will break before too long.
Each shift consists of two parts, into neutral and engage the clutch to match speeds, then disengage the clutch to shift into gear (declutching twice, you see). Going down the box you increase revs in between changes, because the lower gear requires higher engine speeds. Going up, lower revs.
The need is eliminated by synchromesh which does its best to match the speeds before letting you engage gears. Persistent abuse can wear out the synchromesh clutches in the gearbox, though most gearboxes will break if abused that much.