I was curious about this myself up until a few days ago.
To play as the various skins of Batman in the main single player game you have to beat the main story on normal difficulty first. Then when you continue from the main menu you can select what skin you want Batman to wear. You have to select the desired skin each time but that's how it works.
The game restricts you to the default skin while main quest missions are still available so you don't start a pre-rendered cut scene with a different skin from what is presented in the story.
So if you want to free-roam Arkham City with a new skin all you have to do is beat the main story on normal or higher, exit to the main menu, resume your current game, and a menu will appear with all of your DLC skins.
I found the Batman Beyond skin to be very helpful with the AR challenges as the absence of a cape better helps you to judge where Batman's body is at all times.