First, learn how to make SSSR's. It's easy! I recommend Anne's video, or if you have a slow connection, Jane's instructions.
To end with a SSSR, first cut off the other thread.
In this case, the second thread was last used on the second half of a regular split ring. Therefore, it is already knotted and there is no need to tie an additional knot. If I'm cutting off the core thread of a chain, as in Sherry's Treble Clef, which I also end with a SSSR, then I do tie the threads in a square knot before starting the SSSR.
Begin the final ring, tatting over the tail that you just cut off.
When you reach the last stitch of the first half of the ring, continue tatting over the tail for the first half of that stitch (did you get that?) Then bring the tail through to the back of the ring and make the second half of the stitch without tatting over the tail.
Now tat the rest of the ring as you would any other SSSR.
Bring the shuttle through the ring to the back.
Pull the loop to close the ring.
Wrap the loop around and pull the shuttle to tighten the loop.
You can see that (a) this earring needs blocking, and (b) by making a SSSR, you have automatically tatted over the tail of the shuttle thread as you were working. This is the only technique I know of whereby you can tat over the tail of the same shuttle you are working with, while you are working. Cut the threads off and you're done.
Voila, no sewing, no magic threads, it's easy!
The only downside is that this technique can't be used if the final ring is joined to anything else. The join would prevent you from wrapping the loop around the base of the ring. So if the pattern ends with a chain or with a joined ring, you still have to sew in your ends (or use magic threads if you're one of those people who can make that trick work). A lot of Marilee's jewelry patterns end with a ring to attach findings to that is not joined to other elements of the tatting, so these are perfect to finish with a SSSR.
This was one half of one of the pairs of earrings I showed yesterday.