Phonon
Intuitively, there is no reason for lattice vibrations to be quantized.  The macroscopic or large-scale vibrations of a material are not quantized and can take on essentially any frequency.  It is only when we consider the tiny vibrations of the particles that make up a material—atoms and molecules—that the concept of quantization and phonons becomes important.  To understand why phonons exist, complicated quantum mechanics is needed.  Without going into too much detail, the interactions between the constituent atoms or molecules of a crystal are analyzed quantum mechanically.  Because of their arrangement in a pattern that repeats, when one arrives at a mathematical expression for the frequency of oscillation, one finds that there are only distinct quantized values allowed.  There is unfortunately no simple way to explain this.