With bettas, air stones aren't necessary. With other fish, they do become necessary.
Quick explanation: Bettas have their labrynth organ, where they get their air DIRECTLY from the top of the tank (if you get floating plants, as long as there is room to reach air, they'll be just fine - and I do recommend floating plants). The labrynth organ is their very first source of air. Their gills run second, which makes air stones unecessary. If their gills become impaired, they will be fine - they will just come up for air more often. If their labrynth organ becomes impaired, according to this one experiment of removing a betta's labrynth organ, they die of suffocation despite great oxygen levels in the tank. Maybe the gills are simply for not having to go up for air so often. Hope that helped.
Also, with every water change, new fresh oxygen enters the tank.