It starts with one mushroom, which sheds its spores (think of mushroom seeds) around it. The next generation comes up in a small circle around the original spot. Generation 2 sheds its spores, which land in a slightly larger circle. Each generation enlarges the circle until something disrupts the cycle (rain or wind scatters the spores before they sprout). These were once called "faerie rings" because it was believe the mushrooms sprouted where faeries danced.
There are many different varieties of mushrooms but not all mushrooms grow in a "circle" but I think you have seen a patch of meadow mushrooms. They grow in a circle and usually have about 7-20 mushrooms in one patch..