The force which holds the parts of an atom’s nucleus together is extremely powerful but only works for very, very short distances. The protons don’t really want to stay together at all because of charge repulsion (which is pretty powerful when things are that close), so the force keeping them together is extremely high. When you smack a nucleus with a neutron (or a lot of energy somehow), it disrupts that very fine balance of energies, allowing (or forcing) a proton (or many protons) to migrate far enough away from the zone of nuclear force dominance into a region where charge repulsion exerts itself. Often, two smaller nucleii are the result (two new elements are created from the breakup of the one once-larger original one), so it isn’t just one proton flying away, it is a splitting of an atom into two unequal halves. The energy that was keeping the nucleus intact (all the protons and neutrons together) is released. It is a very large amount of energy even from one very tiny event.
The big problem from an energy release standpoint is that the release of energy from one event, and the associated rapid travel of the ejected bits, is enough to cause several other such collision and breakup events to happen, so you get a sudden and roughly simultaneous breakdown of a huge number of nucleii.
There is also a flip side to this process, where we can combine small nucleii together and make bigger molecules. The reason that each method can release a lot of energy is that there is a middle size (the size of an iron atom, which is one reason iron is such a common element) where adding to the nucleus or removing from the nucleus costs more energy than it releases. Elements larger than iron (element #26) release more energy when they break apart whereas elements smaller than iron release energy when combined. Iron sits at the bottom of an energy valley, basically. One side releases energy by combining, and the other side releases energy by splitting.
The breaking apart process is called fission. The adding together process is called fusion. Both require a lot of energy to get started but once they get going, they release more energy than they use to happen, and if the atoms are close enough together, the process feeds on itself, until the atoms get flung apart far enough that collisions with other atoms becomes unlikely. The number of atoms involved is generally HUGE, as is the energy being released by each atom, so KABOOM. It makes chemical reactions seem tame and weak, by comparison (yet chemical reactions are still pretty dangerous things even if not even close to the energy levels released by nuclear reactions).
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