I've seen this idea mentioned as though it were common knowledge, but I've never seen a description of how it would work.

It's no even remotely easy to start up a bank.

I recall from my history of money reading that goldsmiths were amongst the earliest bankers. The reasoning here might be somewhat similar. They were the ones who were accustomed to handling large amounts of gold and storing it securely, so people started paying them to use their storage facilities. Then, they started (over)issuing redemption notes, which circulated far more broadly than the physical gold would have.