What exactly does prolonging the harmony mean? And what limitations does the practice posess?

Hello fellow composers,
Throughout my study of the small musical forms, I have come across some small musical confusion. I am able to create strong basic ideas, however I do not understand how to prolong the harmony underneath the structure of the melody.
While looking at the works of Beethoven and Mozart, I noticed that during their basic ideas they frequently shift from tonic to dominant. When I attempt to do this, my music becomes incomparable. 
Does the harmony have to match up with the notes in the melody? Or can the harmony just shift around on its own while the melody pursues the function of a basic idea? These are a few questions left in my mind after watching the prolongation video.
Thanks!
Dylan