In the website you can browse all the official speedrun categories for Sly Cooper games and their leaderboards. For submitting a run of your own, you need to create an account to the website.
Before submitting your first run, make sure to read both the Game Rules and Category Rules for the category you are planning to run. If you have any questions, feel free to reach out to any moderators or ask in the Sly Cooper Speedrunning Discord!