What is Frappe Mail?

Yes, I used the documentation you shared and I was able to setup everything.

I now need to use the mail server to send emails from other frappe sites. How is that done?

everything = Client + Server + RabbitMQ + Agent(s)?

If yes. Then, create a user in the server, generate the API Key and Secret, and put them in the Client - Mail Client Settings. Follow the docs.