After receiving your payment, KAGI sends you an e-mail with subject "Thanks for your purchase", which contains your registration code.
Then, all you need to do is enter your name and this code into the game and you have the complete version!
Having trouble? Call (510) 658-5244, Monday thru Friday between 9:00 a.m. and 5:00 p.m. Pacific Time. (Store Code: 85N)