If you have purchased an S1, S2 or S3 license, you can unlock LFS using your LFS username and unlock code. You can request a new unlock code on your Personal details page .
On the first screen of LFS after you start it up, there is an "Unlock Live for Speed" button, which takes you to the unlock screen. There, enter your LFS username and your unlock code (in the field named "GAME password") and press the Unlock button at the bottom right of the screen. If you've entered the correct username and unlock code and you are connected to the internet, LFS should unlock.