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 Bis 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.
