You will receive a 30 digit license key with your purchase, which can also be looked up under your account. To activate the program, the key can be entered inside of the Topaz plugin in the bottom Menu -> Enter Key. Make sure to copy and paste your license key in the license key space (not name). You might need to backspace all the way before entering it.
To take yourself off trial mode, simply go to Menu -> Enter Key and enter the key you received for the purchase (there is no difference between the trial software and purchased software, except for the license key).
The most common cause for license key issues is accidentally pasting the key into the "name" field. Please make sure that the key is going into the correct field.
There is a video example attached to this article.