Why does moving air feel cool?

Why does wind feel colder than stagnant air?

On a hot summer day you can sometimes hardly stand the heat. But if you blow the hot air into the room through a fan, it feels colder - why is that?

A freezing cold winter day only stays beautiful when there is no wind, because then the air becomes much, much colder.
The measured temperature of the air is the same whether with or without wind.
The reason why the temperature is felt to be lower when there is wind is the "wind chill factor". The air movement leads to an increase in evaporation on the skin, with every evaporation cold (evaporative cold) occurs, which is then effectively given and felt on the skin as a lower temperature.
The body heat forms a thin, warmed air cushion on the skin. This then protects against the cold. In the wind, the cushion is blown away and the cold air comes directly to the skin. As a result, you feel a strong cooling in winter. This cooling is noticeable even in summer, as the outside air is usually cooler than the body's own air cushion.
Both of the previous answers provide the rationale:

The body hair holds a cushion of air that insulates.
If air strokes along the warmer skin at high speed, it literally transports the heat away, the skin surface takes on air temperature.
Furthermore, the skin surface has moisture and especially when the air is dry, it absorbs water and thus removes heat energy from the skin surface, an effect with every evaporation.
By the way, our body knows this and uses this natural law in the process of sweating when we are too hot.
yes, but jaybeebee's answer is the decisive one, it plays a much bigger role than evaporation.