As can be seen in this question: What effects besides "mass defect" cause the alpha ladder beyond iron-56/nickel-56 to be endothermic? It is not so straightforward to explain why fusion stops at iron (actually it stops at 56Ni and then radioactive decay produces 56Fe afterwards - i.e. 56Ni is the final fusion product, not 56Fe).
Adding alpha particles to 56Ni is still exothermic (in isolation)
The elements around 56Ni, 56Fe etc. occupy a special position, in that they are at the peak of curve of binding energy per nucleon.
What that means is if you have a bunch of nucleons and it is possible to rearrange them to form into nuclei of various kinds, then the natural tendency is to minimise the total energy density by maximising the binding energy per nucleon and forming nuclei that are at the peak of the BE per nucleon curve (i.e. the "iron-peak" elements).
Thus, as highlighted in my answer to the question referred to above, whilst it is energetically favourable (i.e. it releases energy) in a a core made of 52Fe, to break an alpha particle off one nucleus and fuse it with another 52Fe nuclei to make 56Ni, the same is not true if the core is made up of 56Ni. i.e. It consumes energy to break an alpha particle out of a 56Ni nucleus and fuse it with another 56Ni nuclei to make 60Zn.
i.e. There are two steps: (1) Remove an alpha perticle from an existing nucleus; (2) add it to another nucleus to create a heavier nucleus. The two steps taken together are endothermic if the original nucleus is as heavy or heavier than 56Ni. (NB Note that 62Ni is technically the nucleus with maximum BE per nucleon [just], but there is no easy route to get to it by fusion).
Now in the core of a massive star, there is plenty of energy available, so fusion beyond 56Ni could occur (endothermically). However the Coulomb barrier also increases as the nuclei get more protons, and the temperatures required to initiate fusion increase. At the temperatures required to fuse 56Ni to 60Zn then photodisintegration by energetic photons becomes a very important process and so 60Zn (although some is present) tends to be photodisintegrated as quickly as it is produced. Since photodisintegration is very endothermic (see step (1) above), then this usually spells the end of the road for the star and can trigger core collapse.
As for why iron-peak elements are at the peak of the BE per nucleon curve, you need to look at the basic nuclear physics. The strong nuclear force is highly attractive but only operates between nearest neighbours, but nuclei at the "surface" are bound less tightly. There are fewer nucleons at the surface (relative to the total) if you have bigger nuclei. The protons in a nucleus however repel all the other protons in a nucleus; the effect diminishes as the nucleus gets larger, but grows strongly (as $Z^2$) with the number of protons. So you have two competing effects; one favouring larger nuclei, the other disfavouring large nuclei with lots of protons. The binding energy per nucleon is maximised somewhere in the middle.