Cuento literally translates to "story" or "tale," like a children's fairytale. However, in this song, it's used in a very common and clever idiomatic way.
In the lyric "Siempre con el cuento que andabas con tus amigas" (Always with the 'story' that you were with your friends), the singer is sarcastically referring to an excuse or a lie he's heard many times. Learning cuento in this context is great because it captures that feeling of calling someone out on a fib they keep repeating.
This song is a perfect example of someone trying way too hard to prove they're over a breakup. At first, the singer puts on a big show, claiming, "No te preocupes por mí que yo estoy viviendo muy bien" (Don't worry about me, I'm living very well). He says he's forgotten everything and is healing himself with parties and drinking: "Las heridas que dejaste con alcohol me las curé" (The wounds you left, I cured them with alcohol). It's all about looking strong and moving on.
But then, the tough act completely falls apart. He admits the truth: "Me acabé la barra y no te olvido todavía" (I finished the whole bar and I still haven't forgotten you). He's still thinking about her smile and feels like the world is caving in on him. He remembers how he waited for her like a fool, realizing she didn't love him anymore. In the end, he tries to close the chapter for good, telling her to leave him in peace so he can finally heal.