Your engine creates rotational energy. Co-ordinating explosions of gasoline to spin a metal shaft (the crankshaft), your differential performs the important job of helping to transmit that energy to your wheels, but in a very special way.
Imagine you have a barbell with weights on each side. If the weights were glued or welded to the bar, and you had to rotate it. If the weights cannot rotate independently of the bar, the only way to do so is to drag the weights along the floor.
The same is true of your axle and wheels. When your vehicle turns, the wheel on the outside of the turn has to travel a distance further than the one on the inside, meaning it has to turn more. If your wheels cannot turn independently of one another, then the wheel on the outside is forced to drag along the ground until a turn is complete.