All of the transactions are recorded in the blockchain, and verified numerous times. In this case the Exchange states that they were hacked. Which means that the Exchanges' keys were either compromised, or the programming on the Exchange itself was compromised, which in turn would expose the private keys. The exchange of ownership is only done by who has access to the keys. If I tell you my private keys, then we both would be able to transfer BTC to other Public keys. That transfer would show up on the blockchain as well. If I took the BTC and then sold them for fiat(US Dollars), then it's gone as cash.