Slight correction, but that's generally correct. Iron has the lowest binding energy per nucleon, and adding a proton adds the energy to add that proton in to the nucleus as well as energy needed to keep all the existing nucleons together with it, which ends up being more than just adding a proton and keeping all else equal.

The reason for this is quantum mechanical in nature and involves the strong force. It has to do with the strong force being strongest at certain distances before falling off rapidly, and protons don't naturally want to be near each other because they are both positively charged. It also involves spin and nuclear orbitals and other fun things, but the intuitive not really but kind of right idea is that the nucleus is getting large enough that the nucleus is getting large and complex enough that it takes more energy to keep them all together happily and not wanting to change to a new configuration.