The process is pretty simple, and relies on a quirk of iOS: taking a screenshot cancels all touches on the screen. Because of that, anything that you want to protect will require you to have the user to touch the screen to see. If that works for your purposes, the general solution is to simply intercept the touch cancellation, and quickly remove any sensitive information from the screen.
This is great, I had no idea how one would implement screenshot detection, but also haven’t used SnapChat or Facebook Poke so didn’t realize that long-pressing was a requirement. Also I had no idea that Grant Paul uses Tumblr!