Stickiness in rice is also due to what kind of starch is prevalent in the grain. Long and short grain rices have different ratios of several different sorts of starch and this ration also affects how sticky it is.
Evidence of this is the fact that you can have sticky brown rice (short grain) or fluffy brown rice (long grain), even though brown rice is not polished so that the starch is completely exposed.
Anyhow, the fanning is meant to quickly cool down the rice after it is cooked so it retains the correct texture and also (I think) dry out the surface a bit to make it even stickier.
The rice is also supposed to be spread out on a tray (traditionally, in a large, shallow, straight-sided wooden tub) and fanned and turned over with a wooden paddle.