Why does a star die once it has iron?

This is the binding energy of some elements as a function of their number of nucleons

Since fusion leads to a higher number of nucleons, what a star does is to progressively move from left to right on this plot.

So if you start on the left, say H, you merge two, and the result has a larger binding energy, so energy is released. But that behavior is broken when you reach Fe$^{56}$, at that point, you need extra energy. Since the star cannot produce it, it stops fusing material, and it dies off