1. Your transaction goes into their address
2. Their address is always transferring money to accounts.
3. Sometime after you pay them, some amount, not quite the same, leaves their address to an address you control, but which has no established connection to you.
So an observer can see:
1. That you put money into the mixer.
2. The full list of addresses the mixer payed 'out' to (very long).
Which allows them to say if an address has "mixed" money but not to determine which account is connected to which person. If you're careful and you don't transfer any coins to addresses linked to your 'real world' persona, it becomes difficult to trace the account containing the 'mixed' coins to you (though trivial to identify it as coming from the mixer).