A nucleus is not a still, solid ball. The protons and neutrons inside it are in constant motion, jiggling and colliding with each other. This internal motion is a real form of kinetic energy.
Two forces act inside the nucleus. The strong nuclear force attracts nucleons to their close neighbours and holds the nucleus together, but it only reaches across a very short distance. The electrostatic (Coulomb) force makes all the positively charged protons push each other apart, and it reaches right across the whole nucleus.
Because the strong force has to overcome the constant jiggling motion of the nucleons and the Coulomb repulsion of 92 protons, energy is needed to hold the nucleus together. This is the binding energy: the energy that would have to be supplied to pull all the nucleons completely apart.
The same idea works in reverse. When separate nucleons come together to form a nucleus, that binding energy is released. By E = Δmc², the released energy comes from a loss of mass — so the assembled nucleus has slightly less mass than the nucleons it is made of. For U-235 this mass defect is about 1.915 u, equivalent to roughly 1784 MeV of binding energy.
The strong force is short-range, so each nucleon is only pulled by its immediate neighbours — adding more nucleons doesn't increase the pull on any one of them very much. Coulomb repulsion is long-range, so every proton pushes on every other proton across the whole nucleus. In a large nucleus like uranium, the total Coulomb push grows faster than the strong-force glue can keep up.
This is why heavy nuclei are only just barely bound and are prone to instability: the binding energy per nucleon has started to fall (7.59 MeV for U-235, compared with the 8.8 MeV peak at iron-56). A small disturbance — such as absorbing a neutron — can deform the nucleus enough that Coulomb repulsion wins and it splits apart. That is fission, and the energy it releases is exactly the difference in binding energy between the uranium and its more tightly bound fragments.
In this animation, lowering the binding energy lets you watch that tug-of-war tip over: the protons, driven by mutual repulsion, are flung outward first, and the whole nucleus flies apart because there is no longer enough binding energy to hold it against the Coulomb force.