When a root canal [or endodontic treatment] ‘fails’, it means that the treated tooth did not heal–which is the entire point of the root canal treatment in the first place. Root-canal treatment attempt to remove the infection from inside your tooth [fun fact-teeth are not bones, but more like fingernails than bones], and then allow the tooth to heal and return to pain-free function. If the bones around the tooth detect infection [or immunologically smell it, if you will] it will try to deal with it like a sliver in your finger: swelling, drainage [which smells like death when this happens], and pain. If you have a sliver in your finger, and then remove it from your finger, the infection stops, and so does the pain and swelling.
Root canal re-treatment attempts to accomplish the same thing: remove the problem from inside the tooth, and like removing the sliver from the finger causing infection, the tooth will heal. What causes this problem inside root canal treated teeth? Good question: Broken files, missed canals full of dead tissue, and a whole list of other things, but IF root-can retreatment removes the problem, and cleans up the infection, then the tooth can and in many cases heal. It’s also less expensive than replacing the tooth with an implant, bridge or permanent partial denture.