Why is Uranium and Plutonium most commonly used in nuclear fission?

Many contemporary uses of uranium exploit its unique nuclear properties. Uranium-235 has the distinction of being the only naturally occurring fissile isotope. Uranium-238 is both fissionable by fast neutrons, and fertile (capable of being transmuted to fissile plutonium-239 in a nuclear reactor). An artificial fissile isotope, uranium-233, can be produced from natural thorium and is also important in nuclear technology. While uranium-238 has a small probability to fission spontaneously or when bombarded with fast neutrons, the much higher probability of uranium-235 and to a lesser degree uranium-233 to fission when bombarded with slow neutrons generates the heat in nuclear reactors used as a source of power, and provides the fissile material for nuclear weapons. Both uses rely on the ability of uranium to produce a sustained nuclear chain reaction. Depleted uranium (uranium-238) is used in kinetic energy penetrators and armor plating.

The likeliest reasons are its size (number of protons) and will sustain the process by releasing extra neutrons.

To start with, radium cannot be used, it is not fissile.
Uranium 233, 235 and plutonium 239 (Plus a couple of others) are the only fissile isotopes. A fissile material is one which can undergo thermal fission: A slow (thermal) neutron is captured by the nucleus, which becomes a highly excited compound nucleus. Most (not all) of these compound nuclei then split, releasing gamma radiation, fission fragments (with much kinetic energy, i.e. heat) and a number of neutrons which can be slowed down by a moderator to a speed where they can in turn be absorbed by fresh nuclei continuing the reaction.
Uranium 235 is normally used because it occurs in nature in use-able quantities, plutonium is sometimes used in mixed fuel, because although it does not occur in nature, it is a by-product of most reactors as the U-238 in the reactor transmutes due to resonant capture of fast neutrons.
Thorium-232 is also a fertile material, neutron bombardment in a reactor transmutes it to another fissile isotope, U-233 which is also suitable for use in reactors.
The reason fissile materials work is that the energy from absorbing a neutron creates a compound nucleus with enough energy and a too high atomic mass, the compound nucleus is "out of balance" and the binding energy in the nucleus is insufficient to hold it together, so most of the time it fissions. For U-235 about 15% of the time fission does not take place and the atom becomes U-236, which is not fissile.
Adding neutrons to none fissile or fissionable materials does not result in the atoms splitting. However the addition of neutrons "activates" many materials, i.e. makes them radioactive so that they then decay