What does determine the plasma frequency? The formula is:
$$\omega_p² = \frac{4 \pi N e²}{m}$$
and for frequencies above the plasma frequency the material is transparent. So for indium tin oxide (ITO) the frequency of the visible light is above the plasma frequency, which makes ITO transparent in the visible spectrum.
But why? Can the plasma frequency be tuned to a specific value by tuning the charge carrier density N, or is it a fixed value for different materials?
Plasma frequency is generally a fixed value for most materials as it is a function of the square root of $N_0$, the charge density of the material. For most metals $N_0 > 10^{21}$ meaning $w_p$ is above the visible spectrum; this is why most metals reflect visible light (and are shiny). Above $w_p$, the effective permittivity of the material approaches 1 as $w$ approaches $\infty$, meaning the wave travels right through it. If the effective permittivity of a material is 1, it behaves like free space, electrically, which is why the material passes right through it.
The other parameters that define $w_p$ all basically constants referencing whatever charger carrier you're focusing on (usually electrons). We have the charge of the carrier $q$, the effective mass of the carrier, $m_e$, and the permittivity of free space, $e_0$. Can't really change these much.
$w_p = \sqrt{(N_0q^2)/(m_ee_0)} $
