Launch CCC on the unregistered computer. Go to Carbon Copy Cloner > Update CCC... to ensure that you are running the most recent version of CCC.
Click on Check for updates now and if CCC finds one, follow the step by step prompts to allow it to update. Once updated, it will want to quit and relaunch.
Once relaunched, go to Carbon Copy Cloner > Purchase CCC... to check the registration status of the computer.
If CCC is unregistered, you will see the Welcome to Carbon Copy Cloner registration screen. If you have previously purchased CCC, click on Retrieve/Enter Registration Code.
Copy and paste in the name, email address and license key exactly from your registration email. Note: If you try to use a different name or email address, the license key will show as invalid. Click Register.
For your reference, here is what a registration code should look like. Note that the registration code automatically broke between the two lines; I did not use the return key to do so.
Once your copy of CCC is successfully registered, you should see this screen.