[Many](https://en.wikipedia.org/wiki/Beta-Carotene) [dyes](https://en.wikipedia.org/wiki/Quinoline_Yellow_WS) and [pigments](https://en.wikipedia.org/wiki/Azorubine) are [long](https://en.wikipedia.org/wiki/Fast_Green_FCF) [conjugated](https://en.wikipedia.org/wiki/Patent_Blue_V) [organic](https://en.wikipedia.org/wiki/Tartrazine) chemicals which can be broken down by oxidizers such as bleach or hydrogen peroxide. It’s also why [old signs and posters will lose their brilliant reds and yellow](https://www.alamy.com/stock-photo/faded-poster.html)s if you’ve ever noticed convenience stores or hair salons with strangely blue signs. UV degradation from sunlight is another way that these molecules will break down into smaller parts which are either colorless or dull yellowish.
[Bleach](https://en.wikipedia.org/wiki/Sodium_hypochlorite) is an oxidizer; it contains oxygen and chlorine ions that react aggressively with organic compounds and even metals. So basically what happens is the pigments that are used to add color to clothes get oxidized (with the oxygen or with the chlorine) and either lose their color as a result, or detach from the cloth fibers.
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