First of all, you can read about chiropractic activator methods here This appears to be a spring-driven method for delivering a quick, localized “thrust which may or may not “benefit” anything.
Tietze’s syndrome, as described in your article, is about the connective cartilage between the breast end of the ribs (anterior) and the breastbone (sternum). Subluxation essentially means it is partially dislocated and, if you follow the article’s instructions and run your fingers along the sides of your breastbone, you may encounter some discomfort if the cartilage is out of place. (This usually seems to happen with the lower “floating” ribs because of how they connect to each other—rather the sternum).
Cephalad means toward the head (think cephalus) and caudad means toward the tail (think caudal fin in fish) or feet. So the chiropractor is going to feel the joint on inhaling and exhaling to see when the affected joint is stable. Then the chiropractor is going to use his or her hands to push upward or downward to “put things back into place” at the appropriate inhale or exhale as seems appropriate to the injury.
Your article does seem honest in that the chiropractor acknowledged when he or she may have actually caused this condition by a heavy adjustment in the middle of the back (presumably with the patient face down on a table).
Just to be clear though, this is an injury of the mid-chest, not the shoulder cartilage.