Bleedin simple really. The fog actually does move inland, but not necessarily very far.
Sea does not heat up as quickly as land, as anyone who's gone swimming in it knows.
Land warms up, air in contact with land warms up, forms bubbles of warm air that rise up from the surface as thermals. Rising air is replaced by cool air moving in from the sea, what we all know as a sea breeze. See breeze brings the fog with it. Cool moist air meets warm land and starts to warm up, as it warms it rises above dew point and the fog turns to water vapour. The land has cooled down as a result of heat transfer to the cool incoming air and has also been shielded from the sun by the fog so land near the coast gets cooler and the fog front gradually moves further inland as the day progresses.