How it works:
One clutch has 1st gear engaged, with the car moving forwards. Meanwhile, the other clutch has 2nd gear ready to be engaged. Come change-up time, clutch no.1 disengages 1st and clutch no.2 engages 2nd gear and the car keeps going, whilst clutch no.1 prepares 3rd gear for engagement. All of this happens within milliseconds, so power delivery to the road is uninterrupted.
Usually - though not always - dual-clutch gearboxes are more fuel-efficient than an equivalent manual gearbox.
For more info, see Wikipedia.
"Oh, I've got Audi's (VW's) dual-clutch gearbox. It's fantastic!"