As you stand around in the air, you sense the temperature as "cool" or "warm", based on how quickly you lose body heat to the environment (the rate of conduction is proportional to the difference in temperature between two objects). Air is a poor conductor, and without any convection (wind), the thin layer of air surrounding you quickly reaches a temperature that is very close to your body temperature, keeping the heat loss to a low but non-zero level.

Adding wind changes the heat transport situation. A wind convects away the warmed air surrounding the body and replaces it with the cooler, ambient air. This means you have to lose some more body heat to warm the air back up. But this air is also blown away by the wind. Essentially, the body heat loss rate is increased by the convection. You will now sense a new, lower temperature based on this increased heat loss rate. We can determine a windchill equivalent temperature (WET) that would be the temperature of calm air that results in the same amount of body heat loss.

As the wind speed increases, the windchill equivalent temperature decreases for a fixed actual air temperature. While the WET might be chilly to people at many combinations of environmental conditions, the situation becomes harmful to people if the WET drops well below freezing. This is because your health depends on how much body heat you're losing to the environment, which is exactly (and inversely) proportional to the WET, not the actual air temperature.

This is very significant for windy conditions when the actual air temperature is below 0 degrees Celsius; for higher air temperatures, the WET will usually only be low enough to be uncomfortable but not especially unhealthful.