I sent the full balance due to pay off my account, then the bank sent me a bill charging interest. How is this possible?

Even though you paid off your account, there could have been residual interest from previous balances. Residual interest will accrue to an account after the statement date if you have a balance transfer, cash advance balance, or have been carrying a balance from month to month.

From the time your statement cycles until the bank receives your payment, the interest continues to accrue.