As explained on page 15 of the Installation Guide (fsdt_install_guide.pdf, to be found under the fsdreamteam folder under the main flightsim folder), on the paragraph named "Activating your Serial Number", some video cards have issues displaying standard Windows dialog boxes under Full Screen DirectX.
The program hasn't crashed, it's just the registration dialog has gone behind the main FSX screen so you can't see it.
And, as explained on the installation guide, this is usually fixed by putting Flightsim in Windowed mode (pressing ALT+Enter) before the registration. You can then go back to Full Screen after the activation is done.