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...
You will be able to get rid of them - there is a Propietary chemical on that market that will kill fungi - called Armillatox - similar in smell/look to jeyes fluid but is formulated to kill fungi amingst other things eg. algae, slime, lichen and moss.
Is available in garden centres - just dilute it to the correct dose and saturate the ground where the ring is and surrounding area.