Mushrooms disperse their spore in all direction. Then why is there a formation of fairy rings? Why do they grow in one shape and not randomly?
When spores germinate, they do not generate mushrooms directly, but haploid mycelium starts to grow. Only when the mycelium from spores with the other mating type is met, cells fuse and generate diploid mycelium, from which mushrooms can grow.
Fairy rings are caused by a radial outgrowth pattern of the diploid mycelium, followed by resource depletion in the oldest part (middle).
This page seems to have good additional information about hyphal growth.