If there is no bank, who keeps track of who owns what? The answer is a giant shared notebook. Its real name is the blockchain, but "shared notebook" tells you exactly what it is.
Because the notebook is copied onto thousands of computers all over the world. To lie and say "I have more coins than I really do," you would have to change every single copy at the exact same time. That is basically impossible. So everyone can trust the notebook without trusting any one person.
Special computers called miners do the writing. They race to check that everything is honest, and the winner gets to add the next page and earns a little Bitcoin as a thank-you. That is also how brand-new Bitcoins are made.