Why does moving air feel colder?

If the air was still, body heat warms a thin layer of air next to the skin. This warm air would stay near the skin, separating it from the cold air. Wind, however, continuously blows away this warm bit of air, replacing it with the colder surrounding air. There's a similar effect on humidity. Evaporating sweat increases the humidity right next to the skin, decreasing the rate of evaporation. Wind removes this humid air and replaces it with the less humid surrounding air. This is why a fan can cool a person down by blowing hot air at them.

I've also heard stories from soldiers driving tanks in the desert that remaining still can make 120$^\circ$F (49$^\circ$C) days more tolerable. Their bodies create a layer of 98$^\circ$F (37$^\circ$C) air next to their skin.

In addition to Mark H's answer, if your skin is moist the breeze will evaporate water, producing a cooling effect.

As a thumb rule, one feels cold when the skin loses heat to the ambient.

Still air (at temperature lower than that of skin) extracts heat from your skin by free convection between skin surface & near by air molecules this results in rise of temperature of nearby air molecules but when air starts blowing the hot air molecules near your skin is displaced by incoming fresh air molecules. This results in faster rate of heat (forced) convection between your skin and blowing air. This make you feel blowing air colder rather than warmer.

Also the reason why you feel stagnant air hot is that hottest molecules of stagnant air remain in contact with your skin in free convection mode of heat transfer.