They say you can get the whipped cream or egg whites "higher" or "stiffer" if you use a copper bowl. Why is that?
Yes, as mentioned previously it is beneficial to whip egg whites in copper bowls BUT it is important to note that the impact on the egg whites from the copper is primarily beneficial for applications where the final product is going to be baked. You will generally not notice any increased volume in the whipped egg whites themselves, compared to what you'd get without a copper bowl.
As the whites are whipped the copper bonds to create a copper salt that increases the temperature at which the proteins will coagulate. The copper salt makes them more pliable and able to better expand without rupturing. Under "usual" conditions (glass, stainless steel, ceramic) they will coagulate at around 160F degrees. When whipped in a copper bowl they have to reach 170F degrees before they coagulate. This means that they will have a 10 degree increase in temperature to continue to expand and increase in volume.
This also means that if you're talking about whipping egg whites for meringues, dried for cookies or other desserts, the expense of a copper bowl will not be worthwhile as you're not looking for expansion properties in these items. If you're doing a lot of cakes and souffles then a copper bowl would certainly produce better results.
According to wikipedia, the copper bonds to the sulfur in the egg whites, which has the effect of stabilizing the foam.
Cookwise by Shirley Corriher says the same thing.
For whipped cream it helps to have a metal bowl if you cool the cream while you whip by dipping the bottom of the bowl in ice water. You might need to do this if, for example, the cream is warm to begin with.
I couldn't say though if a copper bowl would work better than any other metal bowl.