Partida literally means "game" or "match." While it can refer to a literal game, in this song, it's used metaphorically to describe a significant moment or turning point in a relationship.
Dani Martin sings, "Que me gano la partida" (That won me the game/match), implying that the person's way of being or their influence was so powerful that it 'won' him over, or perhaps changed the course of his emotional 'game.' It's an engaging way to express being deeply affected by someone.
Dani Martín builds this whole song around one stubborn thought: he keeps convincing himself she was the one, then catches himself doing it again. The opening image sets the mood, a girl remembered by her blue skirt and her ever-present cigarette, "Con tu faldita azul y tu cigarro eterno" (With your little blue skirt and your eternal cigarette). It's less about her and more about how specific and unfair memory is, a blue summer next to "un invierno de perros" (a rough, miserable winter), two people who burned out because that's just what they did to each other.
The chorus is the trap he can't get out of: "Cuando pienso que sí que eras tú" (When I think, yeah, it really was you), landing on the title line, "Nada más que tú" (Nothing more than you). He runs through what got him, that half smile, her whole way of being, then admits the fear underneath it all: "Que le apaga la luz" (That turns off the light) on "Al vestido de princesa" (the princess dress), the fantasy fading every time reality shows up. He sings that same chorus three times and never once learns his lesson.