Think of it like email. You need a client (like Gmail or Outlook), which for Matrix is usually Element, Schildichat (a fork of Element), or Fluffychat. You also need a server (like gmail.com). The most popular one is matrix.org, though it doesn’t have any bridges. To get bridges, you either need to run your own server (much easier than it sounds with this) or use a server with bridges built in. Bridges are tied to the server. You also get an address, of the form @name:example.com, where example.com is the homeserver.

If you want to do it the easy (but slightly proprietary) way, Beeper is basically commercialized Matrix with preinstalled bridges and a slightly better UI.