It feels almost intuitive that the moving air would help keep you cool. After all, that's what a breeze does and, in a pinch, waving a folder in front of your face on a hot day will provide a little relief. But since temperature is a feature of the molecular properties of a substance, the air itself isn't made any cooler by movement—it just makes us feel cooler when it blows by.
On a hot day—or on a not so hot day if it's "wind chill" you're talking about—moving air helps your body with the cooling off process. Humans lose heat—a necessity for thermoregulation—through conduction, radiation, convection, and evaporation. The final two are what account for fans' effects. On a hot day, your body sweats to lose heat through the evaporation of that moisture. In still air, that evaporation causes the area immediately surrounding your skin to reach body temperature and 100 percent humidity—rendering it essentially ineffective to continue the process. A fan, or a breeze, helps by replacing this hot, humid air with cooler, drier air that allows for more evaporation.
Similarly, even without sweat, our body loses heat to the surrounding air simply by convection. If our internal temperature is higher than that of the surrounding air, energy—and thus heat—is transferred. However, once again, in motionless air, this simply creates a boundary area of hot air around you. The breeze from the fan carries that hot air away and perpetuates the process, effectively cooling you off.