I haven't tried out hey.com yet. Does it mean I have to fill out some form or captcha in order to send someone an email? Or they would need to "approve" me first?
No. Each time you get an email from a new email address, you screen it and say if you want it in your inbox or somewhere else (spam, or transactional emails, or whatever), or you just don't want to receive it. That way your inbox is only reserved for emails from people you know or accept.