Twitter has a REST API that when requested returns data in JSON format. However, this has little to do with its speed because the multimedia data embedded in the tweets do not come along with the JSON. It's just like the web; how fast the html page has loaded has little to do with how fast the overall page loads in the browser.
The speed you see is because Twitter, like any other big conglomerate, has invested in content delivery networks. Load any Twitter profile in your browser and you'll see several calls made to *.twimg.com servers.
This is where Twitter has all of their static content like images and videos stored. The CDN servers are spread all around the world (at strategic geo locations) to make sure every user experiences as low a latency as possible by fetching these resources from the nearest server node possible.