In my Astronomy class, I learned that temperature results from the speed of air molecules colliding into your skin. Thus, if the air molecules in the room have a high kinetic energy and thus collide with you at high speeds, your temperature will increase. So how does a breeze of wind, in which I imagine air molecules would be moving very fast (or if they're not moving fast, YOU are moving fast relative to them), result in feeling cooler?
Your body is warmer than the surrounding air and as such when heat escapes from your body it warms up that air; if it didn't, you would have overheated years ago. However, if the air isn't moving, that air around you begins to warm up. Heat transfer is faster if the temperature difference is greater. What wind does is move that warm air away, and replace it with other, cooler air. This new patch of air will wick off heat from you more efficiently, which makes you feel cooler. The process is continuous so you start to feel "cold" and hence, wind chill. It's also why blowing on food makes it cooler and easier to eat.
Incidentally, you feel cold because you lose heat, and the faster you lose heat the colder you feel. It's why touching cold metal feels colder than touching wood at the same temperature.