Many pigments are organic compounds, which selectively absorbs some wavelengths of light (thus producing colour). They do so using something called conjugated pi systems. The electrons in this system absorbs the wavelengths of light and become excited.
However, these are susceptible to bond breakage when energetic photons are absorbed - so something like UV light can excite the electron too much and it instead forms a bond with another molecule. This breaks the pi system and the molecule no longer absorbs in the same wavelength. This is observed as colour fading.