Iron has the highest binding energy per nucleon in the entirety of the known elements. But why Iron specifically? What makes it have the highest binding energy per nucleon?
The existence of nuclei is dependent on a number of quantum mechanical boundary conditions. They appear as solutions to a problem where there is a balance of: a) the attractive spill over color force that binds the quarks into a proton or a neutron, b) the repulsive electromagnetic force between protons, c) the Pauli exclusion principle, d) the instability of not strongly bound neutrons to a weak decay. There are additional factors entering once electrons get trapped around a nucleus, but that is another story.
To answer "why" the element with 26 protons and 30 neutrons is stable (or the one with 26 protons and 32 neutrons) and has close to the maximum binding energy, one needs a specific quantum mechanical model for the collective potential of the above factors. Shell models are fairly successful in classifying the periodic table.
The real answer about iron though would be phenomenological, that is what we observe and fit phenomenologically with the Weizsaecker formula, which is based on a liquid drop model. The way the effective potential works, the inclusion of more and more nucleons in the potential well after iron stops creating a deeper effective potential well, due to the increase of the effect of repulsive forces described above.
Please note that it is Ni62 that is more tightly bound in the binding energy curve.
What determines the most stable element (Fe) is the trade off between the nuclear binding (attractive) and the coulomb repulsion between protons. Nucleons feel binding forces that can be described as bulk and surface forces. The bulk forces are those associated with the saturation of nuclear forces (nuclear density in the interior of heavy atoms is relatively constant). Surface terms are associated with the reduced binding at the nuclear surface due to the reduced density. Coulomb repulsion builds up as proton number is increased. It just happens that a transition occurs near Fe (the coulomb repulsion begins to dominate over the enhanced binding due to reducing the surface to volume ratio).