Opting-out puts you in control of how your data is processed. There are 3 different models available depending on how much information you wish to remain in HIBP and how you want yourself and others to be able to search your data in the future. None of these opt-out options affect the notification service; you'll still receive an email if you opted-in to receive them when your address appears in a data breach.
Nobody using the public HIBP search feature will be able to see your email address in search results. You can still search your own address using the notification service that ensures you control the address before seeing the results. The controller of the domain your email address is on will still see you in domain searches.
Your email address is no longer searchable either via the public service or by yourself (even if verifying your address) as the breaches are deleted from the database. Your email address is retained by HIBP to ensure any subsequent breaches it appears in are not added to your record.
The record with your email address is removed completely and neither you nor the public will see it in search results at the time of removal. However, if subsequent data breaches impact that account it will become publicly searchable for new incidents as the opt-out record has been completely deleted.