When air is whisked into egg whites, the mechanical action denatures the proteins in the whites. The denatured proteins coagulate, stiffening the foam and stabilizing the air bubbles. If the foam is overbeaten in a non-copper bowl, eventually the proteins become completely denatured and coagulate into clumps. There is no going back from the clumpy mess to nice foamy whites, so overbeaten whites are usually discarded.
If a copper bowl is used, then fewer protein molecules are free to denature and coagulate, because some are tied up in conalbumin-copper complexes. In addition to forming complexes with conalbumin, the copper may also react with sulfur-containing groups on other proteins, further stabilizing the egg proteins. Although the iron and zinc found in other metal bowls also form complexes with conalbumin, these complexes don't make the foam more stable. When glass or steel bowls are used, cream of tartar may be added to egg whites to stabilize the whites.