The heat radiates out because the clouds aren’t retaining the surface heat. Clouds during the day don’t block all the radiation but do, just as during the night, retain the longer wavelengths such as infrared/heat. They also reflect a lot of radio frequencies for the same reason.
Clouds reflect the surface heat back down, and sunshine back up. It does not matter to the surface what the heat content of a cloud is, only it's thickness and altitude. Blankets don't keep you warm by being warm, but by reflecting your own heat back at you. Due to its inefficiency, a blanket becomes slightly warm from your heat loss, but that has no influence on how warm it keeps you.
Clouds are poor reflectors of radio frequencies. Radar cannot see a cloud, but can see precipitation. On the other hand thick clouds are strong reflectors of infrared heat. Almost all of the heat in a cloud originally came from the surface at the time the water evaporated. Clouds absorb very little new heat from the surface once they are up there.