OpenSSH allows you to transfer files to and from your device using your PC or Mac. You use a SSH client on your computer such as PuTTY on Windows or Cyberduck on OSX, for example. I've used it occasionally to back up Winterboard theme packages and other files, as well as to import custom icon .png's to personalize my device.
It's not really a must have, in my opinion. Another popular jailbreak app that can accomplish the same things is iFile, which is highly recommended. Like using an SSH client, you get full access to the iOS file system right on your device. An actual tutorial on how to use either SSH or iFile is difficult without having a specific task in mind. On top of that, if you don't know what you're doing and aren't careful, it is possible to mess things up. I learned with a lot of help from experienced forum members and a lot of trial and error. There are some how-to threads in the jailbreak section for some specific things like changing the system font. Search through the jailbreak forums for iFile and SSH and you should find some threads that will give you a better idea of what is going on.
To answer your question about iExplorer, it is similar, but with SSH/iFile you have access to the ENTIRE file system. I'm not familiar with iFunBox.