This contraption is a Steenbeck flatbed editor, used to edit motion picture film before it all went digital. The process was unwieldy compared to modern computer editing, but the old methods gave us a hundred years of art.
The table doesn't actually do any editing in and of itself. It merely controls playback. You would shuttle back and forth to the frame around which you wanted to make a cut, then you would place the film into a splicer which would cut the film with a blade and stamp a piece of tape over the seam. So primitive! We were told as film students that editing the old fashioned way at first would force a more sophisticated understanding of editing. Is that true? Does writing on paper force a greater understanding of the written word?
I can't speak to film editing, but as far as sound editing and writing go, I can say that learning it the old-fashioned way first gives a person a great respect for the individual elements of the art. It also helps one to appreciate the ease of more modern methods! Gizmodo has more, including a video of editing film by splicing. Link
(Image credit: Flickr user DRs Kulturarvsprojekt)