The license files (called the Install Key) expire after 14 days.
After 14 days, you need to do a "Restore" to generate a new license file. You can do this all from within Media Center itself. Simply enter your Registration Code (the serial number thing they sent you when you purchased it) into the application. The demo splash thing has a button to do this, or it can be accessed under the Help menu.
It'll register it, and they'll email you a new copy of the license file which can be used on any other Macs that you own.
If that doesn't work for you for some reason (perhaps the computer in question is offline), then you can use this page from any computer to generate the Install Key:
https://rover.jriver.com/cgi-bin/restore.cgiOnce you enter the Registration Code, you can download a new License Key (the file) from the page. They also email it to you whenever you do this. If you can't find your Registration Code (again, the serial number thingy) then you can have them email it to you with that same page.