Exchanges need to know who their customers are to prevent allowing individuals with criminal intentions to use their services. For example there are people joining exchanges and using their services to finance terrorism or for money laundering. If an exchange does not check the background of its users, they risk not-knowingly to enable these people to commit crimes.
The reason why especially cryptocurrency exchanges are an easy target for criminals is that the decentralized nature of cryptocurrency allows transactions to be nearly impossible to trace back to a certain individual. While the information that a transaction happened is stored on the blockchain, it will only lead to a key on the blockchain but not necessarily to the person behind that.
This is why cryptocurrency exchanges have to gather information about the people using the exchange, as anonymity would open the door for crimes.
Especially considering that cryptocurrency can be traded borderless with anyone in the world, not being able to trace people behind transaction would enable money laundering and international conflicts.
