I've got to admit, some of these answers are amongst the most bizarre I come across when describing why you get poor viz under a high!
Basically, a high pressure system (anticyclone) is dominated by slow descent throughout the depth of the troposphere. As air descends it warms due to adiabatic compression, thus the atmospheric column is warmer than normal. Superimposed on this near the Earth's surface lowest 1km or so is the planetary boundary layer that is driven by solar heating that is characterized by buoyancy driven plumes. Under normal (i.e. non anticyclonic conditions) a certain percentage of these plumes are buoyant enough to escape the boundary layer and this results in the 'ventilation' of particulates into the free atmosphere. Also these plumes tend to form clouds which help to 'wash out' particles in the cloud droplets and/or precip.
Anywho, the warmer than normal atmosphere under the anticyclone reduces the buoyancy of boundary layer plumes and reduces the amount of 'ventilation', thus the particulates build up and drive down the viz. There's also some chemical effects from these nasties hanging around that further drives down viz, but that's not really my area.
Hope this helps!
Gareth.