A fan does not make a room cooler by accelerating the average velocity of all molecules in the room. In fact, a fan doesn't directly cool the air at all. The primary mechanism by which a fan creates a cooling effect is through increased evaporation and increased convective heat transfer.
When a fan blows air across your skin, it increases the rate of evaporation of moisture on your skin, which helps to cool you down. As the moisture on your skin evaporates, it absorbs heat energy from your body, resulting in a cooling sensation. This is similar to how sweating cools you down.
Additionally, a fan promotes convective heat transfer. By moving the air around in a room, it helps to reduce the temperature gradient between your body and the surrounding air. This enhances the heat transfer from your body to the air, making you feel cooler.
It's important to note that the fan itself does not change the temperature of the air in the room. It simply moves the air, increasing the effectiveness of natural cooling mechanisms like evaporation and convective heat transfer. If the air in the room is already hot, the fan will not cool it down but can make you feel more comfortable by increasing air movement and facilitating the body's cooling mechanisms.