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.
I kept reading, from what I understood they restored the copy on the original chip, not made a clone.
They could use the backup process to restore a specific partition but then you have a problem of wear due to writes, they've said that a 6 pin passcode is unlikely to be able to be brute forced on a single NAND chip without inducing possible damage due to wear.
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.