I’ve tried looking this up, but all I can find is that they were bankers. And banking simply isn’t profitable if you’re just holding onto money and then giving it back to people. And loans seem like an even worse idea at a time when they couldn’t charge interest.
So I’m confused as to how they actually made the vast sums of money that they did. Has anyone figured this out?
First off, there were people who charged up to 20% interest and were social pariahs for doing so, both Christian and Jewish. They paid a fine of several thousand florins per year in Florence, in effect this was a license for their trade. These were considered to be the lowest of level of bankers throughout Renaissance Italy.
However, about the Medici. The Medici bank's main financial institution was a type of banchi grossi, a Great Bank, and primarily leveraged Bills of Exchange as a way of disguising their violation of Usury. Quidquid sorti accedit, usura est, "Whatever exceeds the Principal is Usury" is the general rule that these prestigious banks followed. There is some complicated ways that the Medici Bank managed to effectively leverage interest in this way in disguise, so I'll give you the gist of it with a common example.
A Wool Trader arrives in the Medici's Florentine branch and pays 10,000 florins with the intention of buying pounds in London in order to purchase the highly desirable English Wool to transport back to Florence. He is given a Bill of Exchange, which is a promise by the London Branch to give him pounds for the florins after a set time period, called an usance. However, where the Medici make money is that the Bill of Exchange instructs the London Branch to pay back the full value in local currency and at the exchange rate set at the start of the usance.
Basically the Medici are betting that the local value of Florins in England will rise relative to the pound by the time the Wool Trader makes it to London. If the Wool Trader borrows Florins at a 1 to 1 exchange rate to Pounds, but by the time the trader gets to London, the value of a Florin doubles, still he will only get back 10,000 Pounds, whereas if he had simply transported 10,000 florins to London himself, he would have been able to get 20,000 pounds by exchanging it then and there. There are some leveraging of the seasonality and locality of certain goods like Wool to increase the chance of a profit for the Medici, but you get the gist of it.
I actually answered a similar question some (gasp!) three years ago, which you can read here, as well as another similar post here. A point which might interest you is that it is in fact a common misconception that the Medici, or any other renaissance lenders, did not or could not charge interest. I discussed that last point here, in another answer about the Medici.