How can I do this “grouping” animation like Apple mail or the Gmail iPad app?
Does this need to be hard coded, and completely done with the animation framework, or does Apple expose an API for this to be done. I checked the developer docs, and I don’t see anything like this. But Gmail was able to exactly replicate Apple’s animation, and that’s why I am curious to know if it can be done in an easier way.

I am 95% sure Apple does not have a UIStackOfPaperViewController or anything like that. You would have to code this up yourself.
Conceptually, it isn’t too complicated: a background view with a texture, and each subview tilted at a random angle and a drop shadow beneath. You could do the drop shadow by drawing a gradient in a view around the “real” view. Core Animation’s default behaviors would probably do the right thing, e.g. if you create the page view off screen and then set the final location, it will fly onto the stack pretty much how you would like it.
If you don’t need support for releases earlier than iOS 6, there’s probably good stuff in the Collection View Controller to make this easier.
Hopefully that’s enough information to get you started, but your question is really broad. It’s difficult to answer “How do I do X?” if you don’t specify your starting point, or answer “How can I do this in an easier way?” if you don’t specify what you think the hard way is first.