How does an oscillating fan work?

It has a mechanism that points it back and forth. The resulting wind increases heat loss and evaporation from your skin, so you feel cooler. The air temp doesn’t actually change much if the air just circulates around. But if you have it blow warm air out of the room, cooler air from the adjoining room will come in. Same if you have it blow cooler air into the room. The warmer air will get displaced into the adjacent room.

More technically, there are several gears in the base of the fan that cause the fan blades to turn and then return; in most fans there are three gears, but sometimes more.

The theory is that consistent fan usage (stationary) will become annoying, while the oscillation provides variety/diversity in what the person feels on their skin.

The science is the same for oscillation or stationary fans: The way a fan keeps you cool is that it either blows cold air across your skin so you lose heat via a process called convection and it helps sweat evaporate faster.

Fans do not cool the air. They move the air causing a human perception of coolness.

The drier the air, the faster the sweat will evaporate from your skin and the cooler you will feel. The fan doesn’t cool the room, it just makes it feel that way. If the humidity of the room was 100%, you wouldn’t feel very cool, if at all, at all.