Integrating 3rd party apps as in being able to use or set 3rd party apps as default selections for files, functions, etc. Like installing a video player from the market and having it become the default player for media files, or open URL links by default with the browser of your choice, etc.
I use my phone to torrent TV episodes that I can watch on the go (rather than be stuck in my DVR at home). It's just another feature that lets Android be an entire computer replacement for me, while with iOS I'm still relying on an actual computer to do certain things.
Gotcha
By Integrated third party apps I think he means that Android apps can share content between each other and the system unlike iOS. For example whenver I take a picture it is automatically uploaded to dropbox without the need to launch any other apps or use a third party camera app.
I have heard good things about the Sharing system in Android, especially ICS.
