The process is simple: Rope light can be cut in sections (there is usually a line or indicator showing where to cut it - it MUST be cut at that point); Cut out the bad sections and either reduce the size of the overall length of rope, or add a new section to make up for the old piece; you will need the rope light couplers now to join the pieces together, here is the problem - finding a coupler and a new piece of rope light that exactly matches and works with your rope could be a nightmare (also, if this is outdoors, be sure to use a coupler rated for outdoor use); join the pieces or add the new pieces according to the couplers instructions (it usually just requires that all pins inside touch the rope light wiring correctly)

If all goes well, you will have rope lights that work again. The other problem you will probably have is that any new lighting will look brighter and whiter than old lighting, especially with incandescent lights. I still highly recommend replacing the entire string (with LED rope, it's much nicer and last much longer)

Good luck!