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.
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OK I think I get it - so the hacker(s) simply "broke in" to the server (this is the breach) - took the key(s) and started to transfer BTC assets from the Bitfinex wallets using the illegally obtained key(s). Is that close to the thing you explained? And therefore, since the correct measures of processing was used (keys) no ripple or trace of false identity is possible. And ownership of any BTC or Satoshi can be changed between to key holders only - and that change is added to the block and therefore verifiable.
A little like if my brother took the key to my safe, takes my 50$ and spend it and I later went to the shop where he spend it to claim it back. And I don't get it back...
Thanks a lot for taking your time explaining the nuts and bolts of the theft!