Why does weather usually become cool and clear after a severe thunderstorm?

Weather becomes cool and clear after a thunderstorm because of the instability in the air along the "frontal boundary" between warm air and cooler air pushing in behind it.

The reason it usually becomes cool after a thunderstorm is because after the storms have settled, the jet stream has pushed the warmer air further east and the cooler air has moved in its place.

Thunderstorms contain updrafts and downdrafts. The downdrafts (caused by falling precipitation) drag cold air down from the top of the storm. When this air hits the ground it spreads out in all directions. The air from this downdraft moving in the direction of the storm is called a "gust front" and arrives just ahead of it and brings a nice cool breeze on a hot day.