Click register serial,
screen go blank then go back to simulator. Zurick still in demo mode.
First, it's best if you put flight sim in Windowed mode before trying.
Also, you don't say if you have seen the dialog that ask you to insert your Serial Number. Assuming you have, if FSX returns to flight without doing anything after you press the OK button, it means you typed the serial wrong.
If you are copy&pasting from email, be sure you don't put extra spaces at the end or at the start of the serial number, which is easy to overlook when copying&pasting from an html email. Also, if you are typing the serial by hand, don't confuse the number "0" with the letter "O"
If the serial has been typed correctly, you should see a question "Do you want to activate Zurich on this PC?". If you don't see the question, then you typed the serial wrong, or added extra spaces at the start or at the end of it.