Why does the addition of oxygen render materials colourless?
Many organic dyes (arguably, all of them) produce color due to electron transitions in conjugated $\pi$ systems. Strong oxidants (not just the peroxide, mind you, but also hypochlorite bleach and others) snap the double bonds and thus destroy these conjugated systems. So the color vanishes.

This is not applicable to inorganic dyes, which usually would not get bleached by peroxide. On the contrary, some transition metals ($\ce{Ti^4+,\;Cr^6+}$, and some others) which have little or no color by themselves would produce intensely colored complexes when hydrogen peroxide is added. These reactions are sometimes used in analytic chemistry to discover their presence.