If you download some free software named PUTTY you can connect to the sever at the command line level and do all kinds of neat stuff! Mostly I watch my son do that stuff because it is linux commands that I have not mastered. If you are old enough to have ever watched the flashing C>: prompt using DOS you will undestand what I am saying.
You can unzip tar and gz files, install software, load databases too large for phpmyadmin. Anything that a tech in the data caenter can do for your account you can do.
With ssh, you can login to your server from your computer. However, the interaction will be based only on the command prompt - so you won't have any graphical applications. In any case, servers don't usually have graphical applications, so ssh is the next best thing to actually sitting near the server with a keyboard, mouse and monitor. You can do everything that can be done with a server using ssh, especially if you are the administrator, with all the permissions to do the different things. If you are just a normal user at the server, you still can manage all your files, apps and stuff.
It's a must if you need to upload a database that is bigger than the php post size and upload limit. It is usually not very high on shared hosts . You can also install perl modules that your host won't want to install. Test shell and perl scripts, etc. Lots of fun and added flexibility to be had.