Here’s how it works: A Canadian lumber company sells boards to a French buyer. The buyer’s bank in France and the seller’s bank in Canada settle the payment, in dollars, via “correspondent banks” that have accounts at the Fed. The money is transferred seamlessly between the banks’ Fed accounts because their status as correspondent banks means they are seen as safe counterparties.
The use of these accounts, the U.S. says, means every transaction technically touches U.S. soil, giving it legal jurisdiction. Because using most other currencies is relatively inconvenient and expensive, many countries and companies will do whatever the U.S. requires to maintain access to dollars.