Metals conduct heat well for two reasons: metal ions pack very closely together in their molecular lattice, and electrons drifting through the metal carry kinetic energy around the lattice. The result is a quick elevation in particle motion that is expressed through heat energy. This conductivity is one reason why one rarely sees metal playground equipment anymore — although the slides go a lot faster than plastic ones, sitting on a slide on a hot summer afternoon is often quite painful.
Within a metallic structure, the positive ions sit close to one another in a geometric layout that is symmetrical. While the ions hold their positions in the lattice, their vibration is constant. Heating the metal causes those ions to vibrate with even more energy. The ions crash into other ions, making their vibration accelerate as well. Passing the energy along in this way is how conducting works, and some of the energy is released as heat.
At the hotter end of a piece of metal, the electrons speed up their movement, gaining kinetic energy from the vibration of the ions colliding with them. Some of them slide away from the heat, running into ions. The collision costs them some of their kinetic energy, but the spreading vibrational energy still ends up leading to the conduction of heat.