I'm sorry the author didn't explain why these words are magical. They flipped the judge (who seems to be hardline) from 180 days jail, to time served. Is there some rule of court procedure that these words invoke? Do the words comprise an implicit threat to appeal?
They don't invoke anything, beyond a simple appeal to humanity. And the fact that they work, when everything else about the legal system is based around mechanical certainty, is terrifying for a lawyer; because it undermines the system they're trained to understand as “if X then Y will (almost) always happen.” If a judge can just decide, “you know what, instead of Y I'll decide to do Σ,” then that shakes the implicit trust of the system. By revealing that the Justice System is based around the whims of a small group of people with disproportionate power.
Username checks out by the way.