It works simillarly to an IRC. You have a server, that server can have channels, I think it can even do voice. But, unlike IRC, you can also use your server to talk to people on other servers, similar to how Fediverse works - if I have a server hosted on myserver.com, and someone else has a public room on server otherserver.com, I can either join the [email protected] or message [email protected], all from my account on myserver.com.

And bridges are basically just bots that run on your own server, and by scraping websites/using API of the service your bridging they create a private room i.e [email protected], with subrooms per chat, and the bot then sends every message it recieves signed into your messenger account to the room, and vice versa - anything you send there will it forward to the real messenger, basically allowing you to chat with people on messenger through your matrix server. Which solves the problem of “Each of my friend is using different messaging service, can I have them all in one app? (The app being Matrix client)”.