Fairy circles consist of isolated or randomly distributed circular areas devoid of any vegetation. They are observed in vast territories in southern Angola, Namibia and South Africa. We report on the formation of fairy circles, and we interpret them as localized structures with a varying plateau size as a function of the aridity. Their stabilization mechanism is attributed to a combined influence of the bistability between the bare state and the uniformly vegetation state, and Lorentzian-like non-local coupling that models the competition between plants. We show how a circular shape is formed, and how the aridity level influences the size of fairy circles. Finally, we show that the proposed mechanism is model-independent.