The first thing I do on any new server is install tmux, and the reason is petty and personal: I am tired of losing work to a flaky connection. You are an hour into a migration, the train goes through a tunnel, the SSH session dies, and the long-running command dies with it because it was tied to your terminal. With tmux, the session keeps running on the server and you just reattach when the signal comes back.
The whole of my daily usage is about four commands. Start a named session so I remember what it was for:
tmux new -s migrate
Detach with Ctrl-b d when I need to wander off, and reattach later, possibly from a different machine entirely:
tmux attach -t migrate
That is genuinely most of the value. The process is parented by tmux, not by my fragile SSH connection, so the connection is now disposable. I can close the lid of the laptop, get on a train, open it on the other end and pick up exactly where I left off, command still running, scrollback intact.
The extras are nice, splits and windows and a status bar I have fiddled with more than I should admit, but they are not the point. The point is that "did my connection just drop?" stopped being a question with consequences. If you only ever learn new, detach and attach, you have captured ninety per cent of why tmux is the first thing on every box I touch.