This transaction is propagated across the network, as any other transaction would be. The mixing node gets introduced to the equation here. The mixing node is in possession of the private key corresponding to the public key that the receiving address was encrypted with. When the mixing node sees such a transaction in a block, it will decrypt it and pull the receiving address from it. It will then create a new transaction to send the (now mixed) coins back to the receiving address.
For privacy purposes, this transaction is not sent out immediately. It is placed in queue, and once a suitable amount of transactions have been created, they will be encrypted with the public key that belongs to the broadcast node. The broadcast node is a node that is geographically separate from the mixing node. It's job, as the name suggests, is to broadcast the transactions created by the mixing node. When the broadcast node receives a group of transactions, it will decrypt them and broadcast them to the network. At this point, that coins have been successfully mixed and returned to their owners.
The mixing node and the broadcast node will both operate behind the TOR network, and be geographically separate from each other. This is done to obscure the location of the master node, so that no entity can determine where it is located. Since the master node is only passively observing block data on the network and sending out encrypted bits of data, it's exposure is minimized. This offers strong protection against the master node from being compromised.
The benefits of this type of system over a traditional mixing service are clear. All possible attack vectors other than the master node being dishonest have been eliminated. Under this system, one must only trust the master node to act honestly, which is required of any system.