Because the exploit pages are lists of leaked passwords, or cc#s. So to check if their password has been leaked they would naturally search for their password.
This is just a very slick implementation of the old "enter your credit card number to see if it's been stolen" technique, except the end-users don't realize that they've given anything away.
Here's an analogy: you get a phone call from someone telling you that your bank account has been hijacked and you need to visit your local branch right away to fix it. So you drive to your bank, enter, and you go talk to a bank employee. As he sits at a computer he asks you for your bank account number, home address, and social security number. After typing them in and hitting a few buttons he tells you that your account has actually not been hijacked and you're safe, so you head home and go watch tv. Meanwhile, what actually happened is that you didn't realize that it was a bank holiday and con-men had broken into the bank and dressed up as bank employees to trick you into giving out your personal information.
I might have fallen for this before. Can't say for sure. Change all the things!
I agree that it does seem a touch unlikely to work in the real world. However, I think a scammer with a touch of ingenuity, creativity, and programming knowledge could probably construct a realistic enough scenario to fool many people.