Amenaces comes from the verb amenazar, meaning "to threaten". It's a strong and direct word that immediately sets the tone for the song.
In "No Me Amenaces," José Alfredo Jiménez uses it repeatedly to confront a lover who is constantly threatening to leave but never actually does. The repetition emphasizes the frustration and emotional manipulation he feels, making it a central and memorable word in the lyrics.
José Alfredo Jiménez sings to a lover who keeps threatening to leave without ever actually going. He opens with the title line itself, “No me amenaces, no me amenaces” [“Don't threaten me, don't threaten me”], and dares her to go if she's really decided to: “Pues agarra tu rumbo y vete” [“Well, take your course and go”]. The joke is in the repetition — “Y te vas, y te vas, y te vas... y no te has ido” [“And you go, and you go, and you go... and you haven't left”] — she keeps announcing her exit and never crosses the door. He treats it like a card game he's already won: “Yo tengo los ases” [“I have the aces”], waiting her out instead of begging her to stay.