You enter the key in the only window that opens when you launch Animation Paper. It’s not on the website. It’s the same window that you must have had open the first time you ran Animation Paper, because you must have clicked the link that is in the same window to register for the activation key.
So in short: Just click your Animation Paper icon. A window opens. That’s it.
Yes, you can’t export or do anything else before you have entered the activation key. It is usually super simple. I have never heard of this problem before. Let me know if the window doesn’t open (for some weird reason).