First, understand that heat is constantly radiating out of your body. This is why wearing clothes helps you stay warm – they trap the heat coming out of your body, and keep it close to you.
The heat radiating out from you radiates into the air around you, heating it up. You can test this yourself – hold your bare hand still in the air in front of you. Now rapidly move it up or down.
See how your skin feels slightly cooler during the movement, from the “wind” of moving your hand?
A fan is essentially doing that to an extreme degree. The air from the fan blows away the little bubble of warm air that hangs around you, helping you cool down in the process.
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