Versus the sun being hidden behind the clouds during a cloudy day
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Because white is the color of all-colors-combined, we literally call light of all colors together “white light”.
You’ve seen how light can hit a prism and separate out into a rainbow, you can also take a rainbow and condense it back into a beam of white light.
In the case of the fog, which is just clouds really, each tiny drop of water is acting as a prism collecting and scattering and randomizing the light into a crazy jumble of all colors together. The result is just a schmeer of white light.
Visible water vapor (like what clouds/fog are made of) scatters scatters only a portion of light that passes through it. Unscattered light carries on and can still form an image of whatever emitted it. The total amount of light scattered is proportional to the distance it travels through visible water vapor. When all the light has scattered it is no longer possible to form an image of the light’s source.
Fog is a thin layer of cloud that forms on the ground. If fog’s thickness is less than the distance at which complete scattering occurs, light from elevated source reaches you before it completely scatters, allowing you to see an image of the source.
Essentially, if you can look up through the fog to see blue sky, then the fog layer is thinner than the scattering distance of its water vapour.
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