I am assuming you are referring to a thunderstorm and not a wind storm.
During storm formation, precipitation forms due to condensation. The water droplets that form from condensation will collide forming larger and larger droplets. In the cases of solid precipitation, a process called the Bergeron process occurs as well as accretion. In either case the precipitation gets larger and larger, until it reaches the point that the gravity acting upon it is greater than the forces that are holding it in the air (updrafts). At this point it falls as rain or snow.
As the precipitation falls, it is removing water from that part of the storm cloud, as well as creating a downdraft. Obviously, since you need water to make cloud, having the water leave the area is going to stop and reverse cloud formation.
The downdrafts caused by falling precipitation is going to work against the upward motion of air preventing addition storm formation.
In the case of a large thunderstorm (supercell), the locations of the downdraft can be separate from the areas of updrafts, thus the downdrafts have almost no effect at all on storm dissipation. In the case of a multi-storm system (several thunderstorms in a row) the downdrafts from the leading thunderstorm can feed into the updrafts of the following storm, so in this situation the downdrafts also do not have much impact on storm dissipation.