SteamChecklist is a website that generates a checklist of the Steam games you own, which you can use to keep track of which games you've played. It can also randomly pick out a game that you still need to play, and it can generate a link to your checklist for you to share.
You aren't giving me your login information! When you press "Sign in through Steam", you're sent to Steam's website (check the URL), and you sign in over there. Once you sign in, Steam redirects you back to SteamChecklist, and sends me your profile ID (and nothing else!), which I then use to grab the list of your Steam games.
If you're already signed in at Steam Community, the "Sign in through Steam" page won't even ask you for your login information. It will only ask you to confirm that it's okay for Steam to send me your profile ID.
It isn't! SteamChecklist doesn't automatically mark your games as completed; you have to do that yourself. There's no way for me to tell when you're done with a game, since it's a completely personal thing. While some games have an achievement for completing the main quest, you may not be done with the game at that point.
To mark games completed, check the checkboxes next to them, then select one of the statuses in the sidebar to the left (Finished, Unfinished, Excluded, etc...).
Click the "Update game list" button. This fetches your current game list from Steam and adds any games you've purchased or started playing since the last update to your checklist.
These statuses are a reference to the reddit comment that inspired SteamChecklist.
In a nutshell, it's a way to methodically work through your backlog without having to power through one game at a time. Pick a game that you really want to play, and mark it as High Interest. Pick another game that seems interesting, and mark it as Some Interest. Finally, pick another game that you have no interest in, and mark it as No Interest. Then, stick to only playing those three games. (SteamChecklist conveniently floats those three games to the top of your checklist.) Once you finish one of them, replace it with another game from that category (e.g. if you finish a game you really wanted to play, pick another game you really want to play).