Clutches are useful in devices with two rotating shafts.
In these devices, one of the shafts is typically driven by a motor or pulley, and the other shaft is driving another device. In a drill for instance, one shaft is driven by a motor, and the other is driving a drill chuck. The clutch connects the two shafts so that they can either be locked together and spin at the same speed, or they can be decoupled and spin at different speeds. In a car, you need a clutch because the engine spins all the time, and the car wheels don't. In order for a car to stop without killing the engine, the wheels need to be disconnected from the engine somehow.