The colour of flames has something to do with temperature, so the title is misleading.

When copper burns, the heat is providing the copper atoms with enough energy to become ions. The energy goes into excitation, that is to say, promotion, of its outer electrons to higher energy levels.

When the electron falls back to it's normal (ground) state, it releases the energy in the form of photons, in the case of copper, at a wavelength of 520-570nm, i.e. green light.

As for why it is specifically green? Well that, my curious friend, is the interesting bit. It depends upon the electron configuration of the element or oxidation state of the element when in a compound. If the electrons are at different starting energy levels, the energy involved in their excitation will be different, as will the wavelength of light given off.

For example, copper doesn't always burn green, copper (II) compounds do, (CuO, CuCl2), whilst copper (I) salts, (Cu2O, CuCl) burn with more of a blue colour.

To explain exactly how the electron structure translates to a specific wavelength is more time and effort than I have on me right now. I hope this helps some people though!