Download the newest version of the program. Install it into your new computer. This will start the trial period of AnyDVD.
Copy the key into your new computer. Double click on the key. You should see a pop up window that says the program is registered, and asks you to restart the program.
If you want to see this in screenshots, Slysoft has a tutorial set up at their forum: http://forum.slysoft.com/showthread.php?t=9284