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If you pay somebody anonymously, how do they know that the payment they've received is yours and not somebody else's? This is something I've always wondered about anonymous payments. I don't see how they can work in practice.

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Several ways to do this, but a simple one: You can generate a unique address where you would know where the payment possibly came from, but nobody looking on-chain could deduce the sender.

To add to the previous response, for your wallet you can create multiple sub-accounts or sub-addresses, that can be used to specifically track payments. You can also use integrated addresses, which tags an encrypted payment id to go with it.

Additionally the sender can prove they sent a specific transaction output to you, without exposing information where it came from.

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