As far as I know, Ihavebeenpwned isn't storing email addresses searched for on their website. I got this from their FAQs section
They can write anything in their FAQ, that doesn't prove anything.
I am a skeptical person by nature and don't believe that someone would provide such service for free.
You may think I am being delusional but here is what I think:
They have a database containing more than 8.500.000.000 email addresses and hashed passwords. Obviously they can't heck them all knowing that most of them are throwaway emails.
By entering your email address there, you are telling them that you care about it and Hence there are high chances that it contains some valuable information.
Also, they provide a service to check if your password was leaked! If you enter your password then they can generate its hash and compare it with other hashes on their database.