This is unlocked by completing the game with any ending, after the last boss you are given an option to "go back in time" or "start a new journey." If you back track in time Extreme mode does not unlock, but once you start a new journey to select Extreme Mode you need to go to the Title Screen, select "options" menu, and change the difficulty from there. You can adjust it back to Normal at any time if it's too difficult for you.