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For the most part, air stones are an esthetic addition to the tank, as many people just like the look of the bubbles rushing upwards. They do provide oxygenation to the water with surface agitation from the bubbles bursting, but most tanks have perfectly adequate oxygenation from the filter.
I like to add them when I am using high temps to treat for ich, for example, just to increase available O2, and they are also helpful in the case of nitrite spikes for the same reason.
They are also used in a fry tank, placed under the eggs, to keep water moving over them and help prevent fungus.