How was South Korean Bitcoin exchange Bithumb hacked? I've read a news article saying that a South Korean Bitcoin exchange had been hacked by someone. My question is: how does this happen? Isn't the Bitcoin network's cryptography safe enough to prevent these events?
The security of Bitcoin is unrelated to someone's Bitcoin wallet being hacked. The network provides the guarantee that given just the data on the blockchain, no one can get your private keys (assuming you haven't exposed them in some way on the blockchain). It does not secure anything related to the environment on which your private keys are stored on. If you were to store your private keys unencrypted on your computer, so long as no one can access your private keys, you would be fine. However your Bitcoin is not safe as if you were to get malware which finds your unencrypted wallet file and sends it to the malware authors, then your coins will be stolen. Your coins being stolen there is related to poor security practices on your computer, not with the Bitcoin network's security.

The same can happen to an exchange. If hackers were to get a backdoor onto the exchange's systems, they could steal the private keys and anything else necessary to access them (e.g. passwords or encryption keys). That the private keys were stolen has nothing to do with the Bitcoin network and everything to do with the exchange poorly securing their private keys.