As I understand it, this happens cryptographically. Send keys can be added to form a larger key, which gets used to sign the pool of transactions. Because the signature used your key as well, the recipient can verify that they have received your coins(from a pool you signed). The important part is that it is impossible to tell who signed what part of the pool, just that one of the people in the pool did. Because all money is pooled together and sent at the same time, it is not possible to read from the amounts sent which transaction belongs to who.