Does anyone else find the opening statement a little misleading? Yes they originally come from one place, but they are sent from Twitter to the app of your choosing via JSON or similar. Sure there's going to be more than one request for the icon sprite and user avatars, but all from Twitter.
"When you open the Twitter app on your smartphone and all those tweets, links, icons, photos, and videos materialize in front of you, they’re not coming from one place. They’re coming from thousands of places."
It sounds like you're misinterpreting. The way I'm reading it, by "thousands of places" they mean "thousands of machines". Which may be an exaggeration, but it should be trivially obvious that data is sharded and must be collated for display.
"When you open the Twitter app on your smartphone and all those tweets, links, icons, photos, and videos materialize in front of you, they’re not coming from one place. They’re coming from thousands of places."