Paul mentioned that they had a way to push new API info to the clients, so I think they basically just re-registered the app as a YOLO thing to see how long until Twitter pulls the plug again.
For that demo, we used the iOS simulator app that's part of Xcode, along with normal screencapture tools (Screenflow in this case). I think that's the way most iOS devs do screencast content, though jailbroken devices can also do native screencapture.
The blog post acknowledges that the inspiration for the app was the Apple site (and a post on HN). None of the code is shared with their implementation (and all Phosphor's javascript is under MIT license, https://github.com/mikewoodworth/phosphorframework), and the Phosphor encoder itself is far more space-efficient than whatever toolset Apple is using internally.