When you visit the site, you just enter the email address you use for various accounts and click 'Check it!' A large database of compromised account passwords and their associated email addresses (see the sources here) will be searched and you'll find out if you're in one of them. If you are, you should change your password right away. If not, you're safe for the time being but you'll be reminded to still change your password regularly.
It's worth noting that Should I Change My Password? isn't a surefire way to know if your accounts are completely safe, but it is a good way to find out if they were compromised. As always, you want to make sure you choose a secure password and avoid putting any sensitive information online if you can.
If you're concerned that the site may just be phishing for emails, the FAQ may make you feel better.
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