First of all, what's a fairy ring? It's a circle of mushrooms which can sometimes surround patches of dead grass inside them. Fairies are said to dance inside the circle of mushrooms, thus giving it its name. However, the explanation for fairy rings is quite simpler and less fantastical. Certain fungi cause the emergence of these fairy rings as they feed on dead organic matter in the soil.
The reason why fairy rings are circular in shape is that the original spore usually starts at the center of the ring, and it spreads horizontally on all sides. Sometimes, mushrooms grow on the edges of the ring, as a part of the fungus. If you wonder whether these fairy rings should be eradicated or not due to their nature of feeding on the grass, fungal experts suggest not to go through the trouble as it would require scooping out the soil where the fairy ring lays. Since it doesn't necessarily harm anything outside its scope, it's best to leave it as is.
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