This is how a mixer works: everyone throws in identical coins and fishes out as many as they threw in, just not their own.
A mixer works like a swimming pool into which many people throw identical coins, and then each fishes out as many as they threw in, only no longer their own. The blockchain shows that you threw ten ether into the pool and that someone fished out ten ether to a new address. These two events cannot be linked if there are enough people in the pool and they throw in identical amounts. This takes cryptography with the funny name “zero-knowledge proof,” which lets you prove you are entitled to take a coin out without saying which coin you threw in.
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