In reply to stucknortherner:
Think of it like a vacuum cleaner and a leaf blower.
Low pressure = Vacuum cleaner, sucks all the crap in.
High pressure = Leaf Blower, pushes all the crap away.
Also as warm moist air rises it cools, reaching the dew point. Water vapour condenses into droplets. Clouds form and it rains. (Low pressure)
Drier air sinking (and warming up as it does) allows for more water vapour to be held in the air and the temperature gets further from dew point so clouds evaporate. (High pressure)
So as there are fewer clouds, the weather is usually clearer during high pressure spells. Also the high pressure areas tend to be a little more stable and can "block" low pressure systems.
Blocking high pressure areas are particularly welcomed by winter climbers as they often mean sustained spells of clear weather which allows the night time temperatures to drop well below freezing.
Overly simplistic but might help.