It starts with a wiki; the company keeps a collaborative site of all the ideas they have for upcoming games, and if the decision is made to go ahead with an idea, that wiki page branches out into the planning documents behind the eventual app. Concepts are drafted in sketches, conversations, and outlines, and eventually they feel like they have the idea "fully developed," at which point the game goes into a proposal pool, and then is eventually picked for production.
That begins with an animatic, as you can see above -- before they ever write the first code line, they map out what will happen in certain game situations. It looks like they used Flash to put that together pretty easily, but you can see how well it shows off game design elements and how things should work during gameplay. After that, there's nothing to it but to do it -- the game is coded, art and sound elements are put together, and of course the usual run of playtesting and so on takes place (the company is going to show off how to test and publish games in a future post).
The game above turned out to be called Symbol6 (Hexago was a working title), and is in the App Store right now. Thanks to Gogogic for sharing a look behind their process -- there are tons of developers working on apps at the moment, and it's neat to see how someone else does it.