Metals conduct heat because they have free electrons in their atoms. When a metal is subject to heat, the free electrons move, spreading the heat to the nearest atoms. The heat is then transferred throughout the metal.
The heat conductivity of a metal is dependent on three factors: how many free electrons a metal has, how fast the electrons can move and how far the electrons can travel before slamming into an obstacle and changing their course.
A metal has a lattice of atoms, each with an outer shell containing electrons that normally dissociate from the parent atom before traveling through the lattice. The dissociated, free group of electrons allows the metal to conduct heat and electric current. When the metal is heated, or an electric voltage is applied, an electric field within the metal triggers the movement of the electrons, making them to shift from one end to another end of the conductor.
The electrons of metals like gold, aluminum, silver and copper can go past hundreds of atoms before bumping into obstacles that make them change their course. This explains why some metals are better conductors of heat than other metals. For instance, in alloy, the electrons only manage to slip through a few atoms before they bump into obstacles.