"Wake up Mr. Wayne, we have much to discuss..."
That line begins a pretty cool intro movie showing you the events leading to the situation at the beginning of the game.
Every inmate from Arkham Asylum and Blackgate Prison has been relocated to a gated area in the middle of the city. That can't be safe for the people of Gotham.
During a press conference where Wayne was announcing that his political plan and his intention of shutting down Arkham City, Bruce Wayne gets captured by Hugo Strange. He threatens to reveal your secret identity if you try to stop him.
The game starts with you sitting in a room handcuffed and tied to a chair. Slowly move the left analog stick (or movement buttons on the PC) left, then right, then left again, keep doing that until you tip over and fall. An alarm will sound and some guards will enter the room.
Follow the on screen commands to counter the attacks of the guard. You should be able to overcome that guard, but then a second guard comes in and since you are handcuffed you dont stand much of a chance. He beats you up and takes you to the courtyard.
Head to Line A as instructed by the guard.
Eventually you run into Jack Ryder, a reporter who was covering the press conference in which you got captured and was taken prisoner also.
A door opens and you enter another courtyard, with prisoners jumping over the fence to attack you. Follow on screen prompts to counter their attacks. Rescue Jack Ryder as well by knocking down his attacker.
More prisoners will attack you, and eventually you will be outnumbered and will get knocked out and lose consciousness.