I was having some difficulty activating my copy of Office 2013 from behind our firewall, and Googled my way to discovering how to install the product key from a command prompt.
For the 32-bit version of Office 2013, cd to c:\Program Files (x86)\Microsoft Office\Office15 and run the following:
cscript OSPP.VBS /inpkey:<your product key>
You'll see some text indicating whether or not the product key installation was successful. The next time you run one of the Office applications, the Activate dialog will appear.
Thanks to the following TechNet article for Office 2010 which pointed me in the right direction.