Visit the bank or open a bank chest to deposit and withdraw. The centered window shows your bank and bag together. Choose 1, 5, 10 or All of that item, then select a bank item to withdraw or a bag item to deposit. Equipped items stay with you. A player-built bank chest accesses your bank; a decorative crate is not a personal inventory chest. The Bank Box spell can call up bank access away from town once you meet its Magic requirement.
Some settlement crates can be searched for small supplies or coins, and some are empty. These differ from decorative player-built crates.