The natural nuclear reactor formed when a uranium-rich mineral deposit became inundated with groundwater that acted as a neutron moderator, and a nuclear chain reaction took place. The heat generated from the nuclear fission caused the groundwater to boil away, which slowed or stopped the reaction. After cooling of the mineral deposit, the water returned and the reaction started again. These fission reactions were sustained for hundreds of thousands of years, until a chain reaction could no longer be supported.
Fission of uranium normally produces five known isotopes of the fission-product gas xenon; all five have been found trapped in the remnants of the natural reactor, in varying concentrations. The concentrations of xenon isotopes, found trapped in mineral formations 2 billion years later, make it possible to calculate the specific time intervals of reactor operation: approximately 30 minutes of criticality followed by 2 hours and 30 minutes of cooling down to complete a 3-hour cycle.
A key factor that made the reaction possible was that, at the time the reactor went critical 1.7 billion years ago, the fissile isotope 235U made up about 3.1% of the natural uranium, which is comparable to the amount used in some of today's reactors. (The remaining 97% was non-fissile 238U.) Because 235U has a shorter half life than 238U, and thus decays more rapidly, the current abundance of 235U in natural uranium is about 0.7%. A natural nuclear reactor is therefore no longer possible on Earth without heavy water or graphite.
The Oklo uranium ore deposits are the only known sites in which natural nuclear reactors existed. Other rich uranium ore bodies would also have had sufficient uranium to support nuclear reactions at that time, but the combination of uranium, water and physical conditions needed to support the chain reaction was unique to the Oklo ore bodies.
Another factor which probably contributed to the start of the Oklo natural nuclear reactor at 2 billion years, rather than earlier, was the increasing oxygen content in the Earth's atmosphere. Uranium is naturally present in the rocks of the earth, and the abundance of fissile 235U was at least 3% or higher at all times prior to reactor startup. Uranium is soluble in water only in the presence of oxygen. Therefore, the rising oxygen levels during the aging of the Earth may have allowed uranium to be dissolved and transported with groundwater to places where a high enough concentration could accumulate to form rich uranium ore bodies. Without the new aerobic environment available on Earth at the time, these concentrations probably could not have taken place.
Here's a photo of the reactor fossil: