What material are the thin, colored wires in my headphone cable?
Yes, all of the conductive stuff inside the cable is copper. However, the copper wires are coated with a very thin insulating layer called enamel.



The enamel can sometimes be removed by rubbing molten solder on it.



If this doesn't work, then you might have some luck rubbing some fine emery paper on it.



As you can see, you don't need to clean your fingernails before you do this.

But one of the problems with headphone cable specifically is that it often contains some thread to reinforce it. 



The thread makes soldering the cable pretty difficult. Usually, if I find threads in my cable, I just throw it away.