Bombachita is the diminutive of bombacha, which can refer to a type of baggy pants worn by gauchos, or more commonly, to underwear or panties.
In the context of the song, it takes on a provocative and unexpected meaning. The lyrics describe Don Zoilo, who becomes a transvestite,
This is a comic gaucho story-song about a marriage that falls apart in a very unexpected way.
The singer marries young, settles into a simple domestic routine, and then his wife leaves him for another man, a flashy gaucho with a knife and a horse. He mopes, plays solitaire, and eventually tracks the couple down at a bar to fight for her back.
The fight turns into something else entirely: staring his rival down mid-duel, the singer falls in love with him instead. The song ends with a wink, the singer's own name is Don Zoilo, and by the final verse he has become "la Zoila," a performer who now travels from town to town, embracing a whole new identity.
It is a playful, self-aware piece of Argentine folk humor: a breakup story that turns its tough-guy setup completely on its head.