Payment by invoice means that when you order a product online, you simply receive an invoice along with the product when it arrives, and you're trusted to pay for the item on the invoice within the next few days or weeks by way of a direct bank transfer.
I found the process quite foreign to me, but the availability of fast and cheap credit checks made this possible. Companies like Klarna maintain and/or license a database of people in the country who have a history of failing to pay their invoices, and they are denied when they order products from other websites in the future.
What I'm wondering here is, how are they going to pull this off in the US? There are credit checks, but they really only apply to whether you've made loan payments or declared bankruptcy recently, and wouldn't reflect if a company has a claim against you.