Well having skimmed through this I dont think anyone got the correct answer (though a few got closer than the rest).
The point is that a high pressure area is created when air descends from above. Air that is above is very dry (because all the moisture fell out when it ascended at some time in the past). Therefore (and this is the important bit) it warms at the Dry Adiabatic Lapse Rate (DALR), so becomes very warm (in relative terms) as it approaches the surface. [The same effect is true of warming winds like the fohn and chinook].
This makes the atmosphere very stable, as it is warm enought to stop most convection from the surface.
During cold nights, surface radation cools the air that is in contact with the ground, leading to the creation of an inversion layer. There may be some vertical movement below the inversion layer but in the main air cannot penetrate through it. The result is that smoke and dust from the surface is trapped in the lowest levels of the atmosphere, with clear conditions above.