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.
Have you ever told an ex you are doing amazing, while secretly drowning in your own feelings? That exact conflict is what Grupo Los de la O and Ivan Cornejo lay out in "Mentiras" (Lies). The track starts with a massive front: the singer insists he feels nothing, claims "las heridas que dejaste con alcohol me las curé" ("the wounds you left, I cured them with alcohol"), and boasts that he is "al cien" ("at one hundred percent" / living at his best) with bottles, live brass bands booked for a month, and "uno de Gelato" (a strain of weed) to erase her completely from his mind.
Then the reality check kicks in. As the chorus hits, all that tough bravado completely falls apart. He admits he drank the entire bar dry ("me acabé la barra") and still cannot shake her smile from his head while feeling like the world is crashing down around him. He looks back at how they fought over silly things ("puras tonterías"), remembering how she always used the excuse of being out with her friends ("siempre con el cuento que andabas con tus amigas"). It is an honest, relatable back-and-forth between acting like you have moved on and knowing you are still hurting.