Essentially, the ground (usually at night) cools down by radiating its daytime heat into the atmosphere. As it gets cooler, it also cools the air directly touching it, to the point where the moisture in the air close to the ground starts to condense into a fog.
The only reason it’s called “radiation” fog is because of the ground radiating its heat off into the atmosphere, the same way anything hot (e.g., pizza fresh out of oven, piping hot cup of coffee) will radiate its heat into the air and cool down. Hope that helps.