He doesn't need to renew them, they are already pointing to the server.
Let's say you buy a domain http://www.cooldomain.com , and host it on digital ocean, then you add that domain to your account (you can add any domain) and then you are told to change the DNS settings for you domain A records to point to DA servers IP.
All good - your domain is now pointing to your DA server IP.
A few months go by, the project doesn't work out and you cancel your server, but your domain is still pointing to that IP address, and remember that there's usually a single IP shared across a ton of users.
So our amigo has an account on one of these hosting providers and finds the domains that are pointing to the IP address that he has. He has no control over the domain, he only needs to add that domain to his account and since it already has the DNS settings pointing to that server - he 'kind of' controls it, hope it makes sense )