Either a log file is written to or a method by which a program/script can call the running transmission-daemon and request a list of all of the torrents which have finished. The daemon will then return a list of all of the torrents that have finished downloading.
This way I can have a script that monitors my download folder making a call every 60min to the daemon requesting a list of all of the finished downloads. Once it has the list it can unpack and move the downloaded files to their relevent directories.