At the simplest level, heat conductivity in metals is governed by two processes - the lattice vibrations and the transport of the free electrons. What makes a metal to be good heat conductors in general is predominantly due to these free electrons. So the rate of thermal conductivity depends on how easily they can move and "carry" the heat from one location to another (since the electron gas has very low specific heat, they absorb heat very easily and also gives off heat very easily).
At "high" temperatures (i.e. room temperature and above), the mechanism that dominates the electron transport in metals is the electron-phonon interaction. Phonons are lattice vibrations (read our FAQ in the General Physics forum if you want to learn more). The coupling between electrons and phonons in a solid is very complicated and depends on, for example, the phonon spectrum. Thus, this can vary from one material to another, and even in a material, can very in one direction versus another.
For copper, it has a relatively weaker electron-phonon coupling than, let's say, Pb and Al. So at a given temperature, the conduction electrons do not get "interfered" with the phonons as strongly as Pb and Al (or even Fe). So it can move more efficiently.
This is also the reason why Cu is also a better electrical conductor than most other metals at room temperature. Ironically, because of its weak coupling with the phonons, it doesn't have enough "glue" to form the necessary Cooper Pairs at very low temperatures and thus, it does not become a superconductor, whereas poorer conductors like Pb and Nb can, due to the stronger electron-phonon coupling strength.
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