How it works, in plain English
Think of it less like Discord and more like email. Nobody owns Matrix. Anyone can run a server, pick a client they like, and talk to anyone else on any other server.
When you sign up here, your address looks like @you:waywardinn.com. Same idea as an email address: it tells you who someone is and where their account lives.
Gmail, Yahoo, a university mail server — they're all different providers, but they can all email each other. Matrix homeservers work the same way. Your account on waywardinn.com can talk to anyone on any other Matrix server.
If a room lives on another server, you can still join it. If someone has an account on a different homeserver, you can still DM them. The server you're on doesn't limit who you can talk to.
When you join a room that lives on another server, our homeserver keeps its own copy of that room's history. This works exactly like email: when someone sends you an email, your provider stores a copy on their servers so you can read it.
This is what Matrix means by federated. Multiple servers all hold pieces of the same conversation.
Discord is one company running one service. If they go down, everyone goes down. If they change the rules, everyone is affected.
Matrix isn't like that. If waywardinn.com went offline tomorrow, the rest of Matrix keeps running. Other servers already have copies of shared rooms. People can move to a different homeserver and carry on. Nothing is riding on any one server staying up.
That's really all there is to it. Pick a homeserver, pick a client, and you're in.