An intuitive explanation why in dielectrics the phase velocity is slowed down, is that the dielectric opposes the field. The dipoles in the dielectric arrange themselves with the positive pole toward the negatively charged plate and with the negative pole toward the positively charged /plate. See what happens with the Coulomb law in a dielectric, the field strength is weaker $ε_r$ times than in vacuum, $| \vec E| = |q| / (4 \pi ε_0 ε_r)$.
As the light is a e/m/ field, what happens with it is similar, the dielectric opposes the electric field. As you can see in phase velocity of light, in materials with relative permittivity $ε_r$ and relative permeability μr, the phase velocity of light becomes $v_p = (ε_0 \mu _0 ε _r \mu _r)^{-1/2}$.As in your case the magnetic permeability is not relevant, $\mu _r = 1$,
$v_p = (ε_0 \mu _0 ε _r)^{-1/2} = c/ \sqrt {ε _r}$.