For a monochromatic plane wave in a transparent nonmagnetic material v = 1/(εμ₀)^½ = c/n.  The index of refraction is n = c(εμ₀)^½.   Since c = 1/(ε₀μ₀)^½,  n = (ε/ε₀)^½ .  For many materials ε depends on ω.  Can we understand how ε(ω) depends on ω?

Let us look at a lih material.
We have P =  ε₀χ_eE_eff = Nα_eE_eff =  Np.
Here χ_e is the electric susceptibility, α_e is the atomic or molecular polarizability, N is the number of atoms or molecules per unit volume, and p is the dipole moment per atom or molecule.  E is the average macroscopic field in the medium.  E_eff is the microscopic field due to everything else except that particular atom or molecule at the location of that atom or molecule.

In a material we often model the electrons bound by a harmonic restoring force to the atomic cores, ie, F_restoring = -kr.   (Our experience tells us that most stable systems that are slightly displaced from equilibrium are acted on by a restoring force proportional to the displacement from equilibrium.  That breaks down for large displacements.)  The "spring constant" k is different for different materials and depends on how strong the electrons are bound to the atomic cores in a particular material.

If we neglect any damping force, then the equation of motion for an electron in the presence of an electric field is
F_total = F_E + F_restoring = ma = md²r/dt², or
F_E = -q_eE_eff = m(d²r/dt² + ω₀²r), where ω₀² = k/m.

If the electric field varies sinusoidally in time and the wavelength is long enough so that spatial variations over the dimensions of an atom can be neglected, we write E_eff = E₀exp(-iωt). The solution to the differential equation then is
r = r₀exp(-iωt), with r₀ = -(q_e/m)E₀/(ω₀² - ω²).

If the atom contains a single electron, then the induced dipole moment of the atom is 
p = -q_er = (q_e²/m)E_eff/(ω₀² - ω²).
If there are f_j electrons per atom with binding frequencies ω_j, then
p = (q_e²/m)E_eff Σ_jf_j/(ω_j² - ω²),  Σ_jf_j = Z.
Here Z is the total number of electrons per atom and the f_j are called the oscillator strengths.

The polarization P is now given by
P = Np = (Nq_e²/m)E_eff Σ_jf_j/(ω_j² - ω²).
But P = ε₀χ_eE_eff , and therefore χ_e(ω) = (Nq_e²/(ε₀m))E_effΣ_jf_j/(ω_j² - ω²) and
n(ω) = (1 + χ_e(ω))^½ = [1 + (Nq_e²/(ε₀m))E_effΣ_jf_j/(ω_j² - ω²)]^½.
If the angular frequency of the effective electric field ω is less than all the ω_j, then n > 1, and if ω is greater than all the ω_j, then < 1.  For visible light in transparent materials n(ω) is, in general, greater than 1, so that v is less than c in the material.

Summary:

Maxwell's equations in macroscopic form predict that the speed of an electromagnetic wave in a medium depends on ε, v = 1/(εμ₀)^½ = c/n.  Measurements show that n depends on ω.  Why?
The susceptibility χ_e depends on the polarizability of the material, i.e. on how the atoms making up the material respond to an applied field.  In our model they respond like driven harmonic oscillators, and the amplitude of a driven oscillator depends on the frequency of the driving force.

Why does an electromagnetic wave slow down in a medium?
Oscillating charges absorb energy from the incident radiation field and re-emit it in the form of a scattered electromagnetic wave.  The apparent reduction in the speed of light in a material is a consequence of the interference of the original wave and many scattered waves all traveling with speed c.