Today, no blockchain
Whenever any of these people want to buy something, their Credit Card connects to their bank/card-issuer's central server. That server accepts or denies the transaction.
This is also where their records/data are stored. If someone 'hacks' or 'leaks' that server the data can be changed.
Crypto Currency, with blockchain
This is how the blockchain works. This customer places a transaction with crypto currency. Instead of one central point that accepts or denies it, a network of thousands of independent machines all check together. When enough machines on the network have confirmed its legitimacy, the transaction is accepted.
This is more secure because it means over 50% of this huge network must be malicious (or 'hacked') in order for it to be wrong; as a hacker, you can't just hack one server!