What does the prefix 'dis' mean in the word 'disengage'?

It means stop, or "un." If you engage in a battle and then your commander asks you to disengage, he is asking you to stop fighting and withdraw. A similar pair would be entangle and disentangle... to stop being wound up in whatever it is. Be careful with prefixes though. It will mean the same thing in a lot of words, but then you will get to something like distance... and there is no such thing as tance, so it doesn't work the same.