It's accurate and not really vague at all, it's just not highschool level physics. I'll try to explain it in layman terms.
Imagine that you have a bunch of free particles that are not connected to each other in any way, and are far enough from each other that any interaction between them (such as electromagnetic fields) is negligible. Ignoring their own masses, the energy of the system is zero. Now imagine these same particles, bound together into a single atom.
Obviously, for the atom to be stable, you don't want it to be able to fall apart on a whim. You want a system where you have to input a lot of energy for the atom to fall apart. But as we just said, the state where the constituent particles are separate is the default, zero-energy state.
Therefore, a stable state where you have to add energy to reach the default free state must actually have negative energy! To be specific, the binding energy is negative while the energy related to the mass of the particles, i.e. e=mc2, is positive. The atom is actually lighter than the sum of its parts!
An atom that doesn't have negative binding energy, i.e. has "excess energy", has nothing binding the constituent particles together, since they have more than enough energy to go run free on their own. Therefore it is unstable.
Elements with a small enough binding energy, small enough that random fluctuations can overcome it and make the atoms fall apart, are what we call radioactive elements.