Why does a rear derailleur need a spring-loaded pulley?

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An African American family of four, all wearing helmets, rides bicycles together on a tree-lined country trail, with two children in front and two adults following behind. — bicycle Family bicycling on a country road.

The rear derailleur requires a spring-loaded pulley primarily to take up chain slack.1

As the derailleur shifts the chain between sprockets of varying sizes, the distance the chain must travel changes. When the chain is moved to a smaller sprocket, the system would otherwise have excess chain length; the spring-loaded pulley maintains the necessary tension to keep the chain drive taut and functioning correctly across all available gear ratios.1