After beating the Champion and after the credits, the Challenge Mode Key is given to you if you are playing Black 2. In order to play Challenge Mode (or Easy Mode on W2,) unfortunately and ridiculously you have to delete your save file, like the Team RR Blaziken said.
The only way to start a new game on challenge mode is to erase all your data, then have someone transfer over challenge mode to your game using their game.
However, you can turn it on as soon as you unlock it, and it will be activated in your game. It won't give a notification or anything when you open the game up, but Pokemon levels will have increased and so on.
Game Freak kinda be dumb
Sources: experience and here