This requires you to save your password in your Google account which sounds like a bad idea from both privacy and security standpoints. Google has proven many times that they can't be trusted with personal data, and you also create more surface for attackers who'd want to compromise your accounts - instead of just one device, all of your devices that are connected to Google could be used to steal your passwords.
They are offering a service to check whether they have leaked your passwords or not, you just have to give them your passwords for them to check.
Makes total sense, as long as they don't leak them yet again

What you want to see at this link is something like this:
