Detergent is mostly soap. Soap binds to water on one end of its molecules and to oil on the other, which allows it to remove oils from fabric and wash them away.
Most detergent also has other chemicals that bind to and react with other dirt related materials but not fabrics and commercial dyes.
By comparison, water just rubs against stuff and abrades the fabric a bit, forcing some of the outermost dirt to loosen and rinse away.
Same goes for washing your hands vs rinsing them.
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