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The inversion traps the air close to the ground and stops it mixing with the upper atmosphere, thus dissipating its heat.
This suggests that warm air is being trapped at low level. In fact, in an inversion there is a temperature increase with altitude.
What happens is that the inversion traps a layer of air below it until the sun's heating is sufficient to warm that air enough to break through the inversion. In a summer high the inversion can be really strong - I've flown on a day when the temperature had to reach 28 degrees before the thermals broke through the inversion.
Air warmed at the ground cools as it rises at about 2.5 degrees per 1,000 ft. So if you have an inversion at, say, 2,000 ft where the air is at 20 degrees, it needs to get to 25 on the ground before convection goes above the inversion and cooler air is drawn down from above the inversion. If it only gets to 24 it stays hot and sticky on the ground.
In winter highs the inversion is weaker, and there is less solar energy to heat the surface (and thus the air). Thus more thermic activity (mixing the warmed air with cooler air from above), and the air is warmed less anyway.