Double-clutching gives you really fast downshifts for overtaking but in reality the technique is only useful for professional racing drivers and truckies.
However, double-clutching will get you a few more months at least out of a manual box that's about to go because the synchros are fucked from years of abuse. It also makes for a smoother shift and reduces wear on your gearbox overall. It's not really necessary though.
It is not quicker to downshift by the double clutching method. The quickest way would be to flatshift or granny shift.
Double clutching is needed in cars without syncromesh in the gearbox.
In the gearbox, there is a shaft that normally spins as fast as the crankshaft in your gearbox, called the layshaft. It connects the driveshaft to the crankshaft.
Double clutching is needed in older cars because when you engage your clutch, the layshaft is disconnected from the crankshaft. However, this is no big deal if you do not change gears, because the wheels keep this shaft moving at a constant rate. However, when you slide the gear into neutral to move it to another gear, the layshaft is disconnected totally, and this reduces the spin. So when you try and move the gearstick into another gear, the layshaft needs to match the speed of the wheels again. But because the layshaft has slowed down, it often grinds and will not go into gear.
Double clutching is where you push the clutch in, put the car into neutral, release the clutch, depress the clutch again, blimp the revs up to above the running revs (the revs your car will be doing in the gear you are changing into), and then guiding (not pushing or forcing) the gear into position when the revs have matched the speed. The blimping of the revs allows the speeds of all the shafts to match each other to reduce wear and prevent grinding.
This sort of technique is not entirely necessary in newer cars, as the syncromesh in most gearboxes brings all the shafts upto identical speeds, taking the need for double clutching away.
