The positive metal ions in a metal structure are packed closely together in a symmetrical geometric arrangement. They don’t move from their position in the lattice but they are constantly vibrating. If a metal is heated, the positive metal ions vibrate more vigorously. These ions collide with neighbouring ions and make them vibrate more vigorously too. In this way, the energy is passed, or conducted, through the metal.
However, metals are particularly good conductors of heat. In general, they are better than ionic compounds which also have strong bonds. So we need another mechanism to explain their especially good conductivity. It is their free electrons.