It's very similar to the way Facebook promotes third-party apps in its app for Android and iPhone.
Developers have the option to plug their apps into Twitter Cards, which show extended content beyond the regular 140 characters of text in a tweet.
For example, if a friend shares a photo from Path on Twitter, you'll have the option to install Path to your own device thanks to a direct link. (If you already have Path or whatever other app installed, Twitter will just launch it instead.)
Before, Twitter Cards were used to show users snippets of articles from links shared on Twitter or photo and video previews.
You can get the new version of Twitter for iPhone and iPad here.