why does the wind or a ceiling fan keep you cold?

Your body temperature (at 98.6°) is usually hotter than the surrounding air, and because of that your body heat creates a thin layer of warmer-than-average air right next to your skin. If the air in the room is still, that layer of warm air stays next to your skin and makes you feel relatively warm. But if the air is moving, the layer of warm air gets blown away, and is replaced by "room-temperature" air, which is usually cooler than the warm layer it replaced. So you feel cooler than you did when the air was still.

All of this happens because we generate our own body heat. If you tried the same experiment with two thermometers in the same room (one in still air, the other in blowing air), you would NOT see a lower reading in the "blowing air" thermometer. In fact, technically, the "blowing air" thermometer should be a tiny bit warmer for the reasons you describe (although the difference would be very, very small unless the wind were blowing very, very fast).

Have you ever wondered why you prespire? It's because when you sweat a water layer is formed on the skin. When air is blown over it the water absorbs heat from our body and evaporates with a cooling effect (Since heat is absorbed from our body ).

When we sit under a fan the air molecules move faster above the skin .

Hence water evaporate faster and we fell coolness . Note when you turn on the fan in a closed room the temperature increases ad you said in the question