I have one of these Speed Brites cleaners and one of the warnings they give about silver chain is that you have to avoid letting the chain bunch together or else you will get these marks, which is I think the electrical current arcing from one piece to the next through the electrolyte cleaner. I am guessing your connection with the alligator clip was not very solid metal-to-metal contact, causing the arcing.
Note that this does not affect karat gold, only silver.
Abrasive polishing is required to remove the discoloration, since it is on the metal itself. In a way, it is almost like firescale, but not as deep into the metal.