Q. What exactly does it mean to be too big to fail, anyway?
So: Banks are basically in the business of owing money to people.* All the money in your checking account is money the bank owes you. And banks' trading businesses are mostly conducted with borrowed money.**
Q. So why would a bank want to be too big to fail?
Remember: banks are in the business of owing people money. Being too big to fail makes you better at owing people money: if you somehow lose all their money, the government will step in and give you money to give back to them.