You probably won't... fairy rings are caused by fungus similar to toadstools or mushrooms and are, themselves caused by an abundance of organic matter below the soil surface. You may have had a tree or somethind die and left the stump rotting away... at any rate... you could try digging it up, but my experience is to just leave them alone...I do fertilize the grass a little extra in the area to darken the color of all the grass which makes them somewhat less noticeable...