Why does larger permittivity of a medium cause light to propagate slower?

I was wondering about what physically happens when light is transmitted through a non-magnetic medium. Specifically, I’m trying to visualize how materials slow down light as the electromagnetic wave is passing through, and how permittivity affects this. I know that the index of refraction is directly related to relative permittivity, but I’m unclear as to how this parameter affects the speed of propagation.

My understanding of permittivity is that it measures how easily the molecules of the medium can polarize due to the electric field component of light, with larger permittivity meaning easier polarization of the dipole moments. These polarized molecules in turn have a growing/shrinking electric field between the poles that eventually counteracts the initial field that polarized them.

I’m thinking that this time-varying electric field creates a magnetic field, which then creates an electric field, which then creates a magnetic field and so on, and the speed of the light traveling through the medium is dependent on how quickly these fields rise and collapse. This would suggest that my interpretation of a larger permittivity would cause faster propagation, but I know from the equation that a larger permittivity means a larger index of refraction and a slower propagation of light.

My reasoning is flawed, but I’m not sure where I went wrong. I'm thinking that my understanding of permittivity is incorrect. I was hoping someone could shed some light on what physically happens as the waves propagate through a medium, and how this relates to permittivity. If you have any suggestions on websites or links I should look at it would also be greatly appreciated.

If I can expand a little bit on Sofia's answer the polarization of the medium opposes time variations in the electric field thus slowing down the phase velocity of the wave.

This can be seen from Ampere's circuit law (the 4th Maxwell equation) which is central as you stated in arriving at the wave equation describing light. It can be written in vacuum as

$\frac{\partial \mathbf{E}} {\partial t} = \frac{1}{\varepsilon_0\mu_0}\nabla \times \mathbf{B}$.

It says that physically the coupling between the time variation of E and the curl of B is inversely proportional to the vacuum permittivity making it plausible that a larger vacuum permittivity would give a lower phase velocity of the E wave.

To be completely rigorous one still would need to solve the coupled Maxwell equations in the usual way leading to the usual expression of $c$ in terms of $\epsilon_0$ and $\mu_0$ but I believe this gives an argument.

This can be easily extended to say a isotropic medium in which the medium polarization works in the same way as increasing the vacuum permittivity. In short, in a medium with permittivity > 1 the polarization opposes the rate in which the magnetic field causes the electric field to change over time.

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, $\mu _r = 1$,

$v_p = (ε_0 \mu _0 ε _r)^{-1/2} = c/ \sqrt {ε _r}$.

Yet another way, in terms of an ensemble of oscillating charges, to see this: the overall effect of incident light is to induce the charges to oscillate with a phase delay (because they react) which when added up turn out (Ewald–Oseen extinction theorem) to be somehow a polarization field with shorter wavelength but same frequency, as explained here.