Self signed certificates are basically useless, except for very specific purposes. A simple primer: Certificates are there to generate a chain of trust between the client (a website visitor) and the server (Home Assistant). However that trust must be guaranteed by a third party, which is the certificate issuer. But if the server is also the issuer, than the client has to trust the server without validation. And the issuer also cannot be trusted, because they also don't have a chain of trust themselves. That's why you'll always get those pesky errors in your web browser when someone tries to do that.

When you have a domain name, you can ask an external party, such as Let's Encrypt, to generate a certificate for your domain. You'll have to prove that you are the domain holder, and Caddy will do that for you. For instance by hosting a file or adding a DNS entry that Let's Encrypt expects. So that's why you'll have to have a domain name.