MIT explains phonons. To understand how heat spreads through a material, consider that heat — as well as sound — is actually the motion or vibration of atoms and molecules:
Low-frequency vibrations correspond to sound, while higher frequencies correspond to heat. At each frequency, quantum mechanics principles dictate that the vibrational energy must be a multiple of a basic amount of energy, called a quantum, that is proportional to the frequency. Physicists call these basic levels of energy phonons.
In a sense, then, “phonon” is just a fancy word for a particle of heat.
Phonons are especially relevant in the behavior of heat and sound in crystals