It's not a UINavigationController replacement. It's a slide menu like facebook, just uses nav controller as inspiration, to make creating slide menus, that are all the rage, easier to setup use and dictate what goes on your navigation list. I will work on a better screen shot, but there isn't much for a screen shot to show. It also uses iOS containment as well.

Looks like all the view controllers have to be loaded at start up, unless I'm reading the code too hastily.. Seems to me it would be better to load them up on demand

You can load them on demand as well, pushViewController when ever you would like.

There's no motivation here to why I, as an iOS developer, would want to use this over UINavigationController. What makes it better? What problem does it solve? The one screenshot it has looks terrible.

I agree the screen shot he used is terrible. Take a look at Path if you have the app. When you tap the Add Friends button in the top right the main view controller slides over to the left, but not all the way. About 20 px remain on the screen. It's a nice effect becaause it gives the user a feeling that they are still "on" the main screen, just tweaking some settings in this underlying part. The transition to settings is less jarring.

There's also https://github.com/pkluz/ZUUIRevealController which implements iOS5 containment, and i think they ported it over to ios4