To clarify, the splitting of a single atom does not produce a nuclear explosion on the scale of the bombs dropped on Hiroshima and Nagasaki. The magnitude of the nuclear explosions in those instances was the result of the simultaneous fission (splitting) of thousands of atoms within the nuclear material.
In nuclear weapons, such as the ones used in Hiroshima and Nagasaki, a chain reaction is initiated by bombarding a critical mass of fissile material, typically enriched uranium or plutonium, with neutrons. When an atom of the fissile material absorbs a neutron, it becomes unstable and undergoes nuclear fission, splitting into two smaller nuclei and releasing a large amount of energy in the process.
The energy released from each fission event is relatively small, but the key is that each fission event releases additional neutrons, which can then go on to cause more fission events in nearby atoms. This creates a self-sustaining chain reaction, leading to a rapid release of an enormous amount of energy.
In the case of the Hiroshima and Nagasaki bombs, the nuclear reactions occurred in a highly compressed and critical mass of fissile material, resulting in a catastrophic release of energy. The splitting of thousands of atoms in the nuclear material within a fraction of a second generated an explosion equivalent to many thousands of tons of conventional explosives.
Therefore, it's the cumulative effect of the fission reactions occurring in large numbers of atoms within the nuclear material that produces the tremendous destructive power of a nuclear explosion.