When the frequency is increased the permittivity of a dielectric material decreases. But in the way of decrease there are some spikes which indicates sudden increase of permittivity for certain frequencies. What are the reasons behind it?
This peak is typical for the resonance curve of any (weakly damped) harmonic oscillator. It naturally comes as a solution of its differential equation for different frequencies.
Driving the oscillator at a frequency slightly below its resonance frequency leads to big amplitude of oscillations. Therefore, even a weak electric field causes a strong electric polarisation.
The dielectric function of crystals can be approximated a sum of multiple independent Lorentz oscillators, each corresponding to some type of vibration of atoms in the lattice or of their electronic shells. Your plot would be better also accompanied by the imaginary part of permittivity, which exhibits peaks nearly centered at the resonance frequencies.