Automatic wrist watches, like any other watch, also need energy for functioning. This comes from the up, down and twisting movement of our hand.
There is a small pendulum-like component which rotates around a fulcrum due to our hand’s movement. The movement of the pendulum (clockwise and anti-clockwise) is used to wind a spiral spring. This action is similar to winding it manually. The energy stored in a fully wound spring is adequate to run the watch for a day or two. Within this time, if the watch is not worn, there is no recharging of the spiral spring. Hence the watch will naturally come to a stop.
Unlike winding type watches, automatic watches have a different mechanism: a rotor which can rotate freely (like a free wheel in a bicycle).
Our hand movements make the rotor rotate and wind a coil spring. The rotor frees itself to the original position after the winding. As a result the rotations of the rotor further tighten the spring. The watch works as the spring releases very slowly.
There is also a release mechanism by which the rotor is freed completely when the coil spring is wound to a preset maximum.
If the watch (spring) is kept idle for some time, there is no rewinding of the spring, and so it stops functioning.