Offline mode only works with the desktop version of Playground. iPads need to be connected to the internet at all times. See my two screenshots for information on how to log into the desktop app while offline:
This is the first thing you'll see when you launch Playground while you're not connected to the internet. I had to click the "X" rather than continue to get to the next screen. Try doing that.
Then, you should see this option menu. Proceed to whatever you'd like. Remember, you have to have previously downloaded lessons in order to use them offline.
General user help and bug reports for Playground Sessions for Windows