When one end of the solid rod gains heat, the heat energy absorbed is converted to kinetic energy in the rod particles. The particles at this end of the rod vibrated more vigorously in their fixed positions. The increased vibration is passed on to the neighbouring particles, causing the neighbouring particles to also vibrate more vigorously. Hence the increased vibration of the particles is passed on from particle to particle, till the particles at the other end of the rod also vibrate more vigorously.
When the average kinetic energy of the particles in the rod increases, the temperature of the rod increases. Heat energy is hence transferred from the hotter end of the rod to the cooler end of the rod through vibration of particles.
In metals, besides particle vibration, there is also electron diffusion to transfer the heat energy. The delocalised electrons are able to move more freely, hence heat conduction in metals is faster compared to non-metals.