ELI5: How can banks lend out far more money than they physically have sitting in their vaults?

Banks can have enormous amounts of money loaned out through mortgages, business loans, credit cards, and other lending, even though they obviously don't keep an equivalent amount of cash sitting around.

What actually happens inside the banking system when a bank gives someone a loan? Where does that spendable money come from, and what prevents banks from creating unlimited amounts of it?