The first observation of isolated attosecond pulses by Hentschel [Nature 414, 509 (2001)] resulted from an experiment that left the exact mechanism of their generation unresolved. A complete simulation of the experiment reveals the reason for its success: single pulses were efficiently isolated from two or more generated pulses by spatial filtering in the far field. Our explanation suggests a new, simple paradigm for the production of isolated attosecond bursts. We show that this method can be used, in conjunction with carrier-envelope phase stabilization, to select single attosecond pulses by use of 10 fs driving pulses.