It's a "terminal multiplexer", i.e. you can start multiple terminals in a single terminal.

You might ask, why not open a new terminal window or tab? Well, you can only do that in a desktop environment and that's not always available. Even if you can, you might want the terminals to be side by side in a single screen, which might not be easy to do with window tiling.

The real power of tmux, though, is that it manages the session you created. To quote from the manual:

tmux may be detached from a screen and continue running in the background, then later reattached.

So, one use case would be saving your current terminal setup. Instead of exiting the terminal and navigating to the project and setting up the environment again next time, you can simply detach and re-attach.

When connecting to a remote server, this is especially useful:

Each session is persistent and will survive accidental disconnection (such as ssh(1) connection timeout) or intentional detaching

Suppose you want to execute a long running command on a remote server. If you just put it to foreground, when you exit the ssh session, the job is also killed. If you put it to the background, its output can't be easily observed.

With tmux, you can simply run it in the foreground like normal and detach. When you reattach later, the job is running and you get all the output easily, as if you have been in that session all along.