Rompecabezas literally translates to "head-breaker", but it's the common Spanish word for "puzzle". It's a fun and evocative word that paints a clear picture.
In the song, the singer uses this word metaphorically to describe himself as a piece in his partner's puzzle that "nunca hizo falta, que no encajará" (was never needed, that won't fit). This imagery powerfully conveys feelings of being unwanted and incompatible in the relationship.
Gilberto Santa Rosa breaks a relationship down into arithmetic, and the arithmetic is the whole conceit: "Ahora solo hay símbolos de suma y resta" [Now there are only plus and minus symbols]. The sums run one way only — "Sumas mis errores, resto tu bondad" [You add up my mistakes, I subtract your kindness] — and he has already worked out what he amounts to in them: "Ahora soy la pieza en tu rompecabezas / Que nunca hizo falta, que no encajará" [Now I'm the piece in your puzzle that was never needed, that won't fit].
Then he counts, ten down to one, pinning a separate complaint to each number. The sharpest of them is a pun rather than an accusation: "Siete, casi siempre llego a las / Seis" [Seven, I almost always get in at six] — the next number in the countdown doubles as the hour he comes home, so the grievance and the counting are the same sentence. By "Dos, si no hay amor, no hay nada" [Two, if there's no love, there's nothing] the argument is finished, and the song ends on the single word the counting was always heading for.