How do race drivers clear foggy windscreens?

Fogging happens when there's a temperature differential between one side of the glass and the other. Most racecars are open - they remove the windows to save weight too, so the temperature inside the car is the same as the outside. As there's no difference in temperature there's usually no mechanism for fog to form in the first place.

Still, the conditions can sometimes still happen with rain or very high humidity. In those cases a simple blower system will work okay, there are also spray or wipe on chemical coatings you can put on to prevent fogging. In the X-1 program that broke the aviation sound barrier in a rocket airplane they wiped on Preen shampoo believe it or not! An AC compressor certainly helps defogging but it's not essential.

Fog on your windshield doesn't occur due to the difference between one side of the glass and the other, it's because of the dew point of the air. If the glass is below the dew point of the air it comes in contact with (whether inside or out), fog will occur on there unless there's no moisture in the air (which effectively lowers the dew point). You usually fog up inside your car because your breath is hot/moist enough to cause the dew point to come into effect. Race cars don't have that problem because drivers have helmets on which prevents their breath from reaching the windshield, that coupled with the fact there is air flowing through the inside of the race vehicle to help the driver stay cool. I don't know for sure, but would assume race teams coat the windshields with whatever they need to to prevent it from fogging up. They wouldn't leave something like this to chance as a fogged windshield could lose the race for an entire team.

It's not usually a problem. The inside of most competition vehicles is pretty warm, so once you've cleared off any misting in the morning (using a cloth usually), it doesn't tend to fog up again.

Saying that, we used to use either 'rain-x' or toothpaste rubbed on the inside of the screen to stop it misting on colder night rallies.

AC certainly isn't necessary, that's a relatively recent invention which few cars had before the mid 90s...