Marchar means "to leave" or "to depart." While it can be used in everyday contexts, in this song, it carries a strong emotional weight, making it more impactful than a simpler word like "ir."
In "No Te Vayas por Favor," the singer repeatedly pleads, "¿Te quieres marchar?" (Do you want to leave?), emphasizing the pain and desperation of being abandoned. The word highlights the central conflict of the song: the fear of a loved one walking away.