You want to start the process by going into the 'Register' menu and entering your serial number and press the OK button. You will get a window titled 'Activation Options'. Here you want to choose to 'activate using a different computer that has web access'. Click the next button and then the save button to save the activation URL and installation ID to a text file so you can take it to a computer where you have internet access.
On that computer, go to the activation URL and enter your installation ID into the field provided. That'll give you an activation key which you can then enter back on your home computer into the fields provided when you choose the option 'I already have an activation key and would like to activate it now' under that same 'Activation Options' window that you encountered above.