The vibration is transmitted or transferred to the neighboring atoms or molecules. More electrons are released which move towards the colder end. Thus the heat is conducted . Conduction of heat is defined as the mode of transmission of heat, through a medium, without actual migration of the atoms or molecules of the medium. For conduction of heat, a material medium is necessary.
Conduction occurs mostly in solids.
This can be understood if we see how solids, liquids and gases are bound to their respective atoms or molecules. Solids are tightly bound. The atoms or molecules cannot migrate from one site of the solid to the other. Although, the electrons can do so. In liquids and gases, the atoms or molecules are not that tightly bound, so when the heat has to be transferred or transmitted, the atoms and molecules themselves move.