Atoms split in atomic bombs when their physics package (the fissionable material like enriched uranium or plutonium) is flooded with neutrons at just the right velocities. If the neutrons are going too fast, they simply rebound off the atoms. If they are going too slow, they are not captured. Like in the three bears, Goldilocks has to have it just right.
Unlike what one answer said, going very very fast does not do it. This results because the neutrons do NOT split atoms so much as the atoms fly apart because their energy levels were raised by captured neutrons.
And that's it, when atoms have too many neutrons, they fly apart into children parts whose sum of masses is less than that of the original atom. And that mass difference delM gives us E = delM c^2. That difference in mass is where the energy for atomic bombs comes from.
The amount of energy produced by each atom flying apart is very small...hardly the megatons of an atomic bomb. But when you take in billions and billions of atoms coming apart within less than a second, that adds up to the big bang. And we get billions of these uranium or plutonium atoms going bust because for each atom splitting, more than one additional neutron is produced to split even more atoms. That's called a chain reaction. KABOOM.