Each institution printed these postcards for its researchers. They included the institution address and a template request. To request a reprint, you would fill in the address of the author and some information about the paper you were interested in and drop it in mail. And hope for a response in six to ten weeks. Here are a couple of requests that my uncle received.
Much has changed since those days. Journal archives have moved online and email zips across the world in seconds. It is hard to imagine today how researchers of the day moved the mountains that they did.