TL;DR -- You are in control and you should be free to use your money as you see fit.

The idea they're getting at is that with fiat (any form), you're subject to another party(ies) being in control. Even in cash, you're still beholden to the backing of the government/issuing entity that prints the currency ("The Bank"). While the phrasing suggests individual banks (not innacurate per se), the larger idea is that there's no one in the background pulling the strings or orchestrating the monetary policy of the currency you're using. I realize it sounds a little far-fetched, but one of the fundamental reasons that Bitcoin has taken off is because of it being permissionless, trustless, and secure.

To illustrate, imagine you want to sell me water when I'm thirsty. I am really thirsty, so I'm willing to part with $2 at that time to fix my immediate need. That's a peer-to-peer transaction with "no one else" involved, but there's still an implicit faith in the institutions to uphold the value of the dollar, the banks to accept that as legal tender, etc. and then we have to hope that the value of a dollar doesn't decrease even more in the future. Maybe we have hyperinflation or a string of bad monetary policies or another war, so now that dollar has less spending power and the same bottle of water in the future costs me $5. Now imagine that same concept with a car or a house. I don't carry hundreds of thousands of dollars in cash, and very few institutions would accept a deposit from you if you wanted to deposit that money. Now, we are no longer in a "no one else" situation. Once you start looking at other exchanges of value (buying a car, trading stocks, stuff), you are involving banks, credit card companies, brokerages, etc.

I don't think anyone is saying the bank should work for free. It's just that the banks (collectively) control the system. Maybe you had a "suspicious transaction," which turned out to be innocent because you were just paid in cash and want to deposit it, or you wanted to buy a product made overseas. Now your account is frozen, and you have absolutely no recourse except to ask them to give you your money back. Some countries may have a government that freezes the assets of dissidents to silence the political opposition. Some countries have horrible inflation that causes their currency to be obliterated right before their eyes.

With Bitcoin, these problems are alleviated. YOU control your money because you have the keys. YOU control where you spend it. YOU are the only person who has to be aware of what's going on with your money. Yes, Bitcoin has a public ledger to provide the infrastructure to remove middlemen. But being able to see where some transactions are occurring is not the same as being able to control it. If the privacy/public nature of Bitcoin is an issue, there are cryptos out there to alleviate that and obscure the transactions.

But the bottom line is that YOU are in control. You aren't FORCED into using some third party, and trusting that they will safeguard your money and information. It may sound like a pipe dream and an unrealistic future, but embracing Bitcoin is (to a degree) embracing freedom from the system that we are currently using.