What happens in an anticyclone?

The wind circulation around a high is called anticyclonic circulation. Around a high, it is called anticyclonic circulation. An anticyclone is an area of high pressure where air moves apart and sinks. It typically indicates fair weather. Winds in an anticyclone blow clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere. 

Air at the centre of an anticyclone is forced away from its area of high pressure and replaced by a downward blast of air from higher altitudes. The air compresses and heats up as it moves downward, reducing its humidity and leading to fewer clouds within the anticyclone.

Further Reading:

• General Circulation Of The Atmosphere
• Tropical Cyclones