Scimmia literally means "monkey". However, in Italian slang, "avere la scimmia" (to have the monkey) is an idiomatic expression that refers to having an intense craving or addiction, often for drugs, but it can also be used for other strong desires.
In this song, Fabri Fibra uses it to express his intense drive and almost addictive need to rap, saying "Mi sale la scimmia per ogni rapper che mi cita" (The monkey climbs on me / I get a craving for every rapper who mentions me). It's a vivid and memorable way to convey his passion and competitive spirit in the rap scene.
Fabri Fibra spends this song measuring the distance between where he started and where he ended up, and finding that the distance is smaller than people assume.
The picture he keeps returning to is the demo tape. Before any of the money, he was writing verses onto cassettes and posting them to every DJ in Italy, then hearing his own demo come back at him through a stereo. That memory is the chorus, and it works as a reminder rather than as nostalgia. The line he repeats around it is the point of the whole track: he made money, but he stayed the same person. Only the fee changed, not the team around him.
The verses are a tour of the Italian rap scene as he sees it now. He counts off how half of Milan copies one rapper and half copies another while the audience happily takes more of the same. He notes that fame behaves like someone who looks you over and tells you that you are not her type. He is dismissive of rivals, protective of the people who were there early, and openly amused by the envy that arrives with every banknote.
Underneath the boasting there is a quieter argument about staying recognisable to yourself. He admits how much time the sheets of paper cost him, keeps his accent even after moving north, and insists that the real ones still recognise each other. The demo in the stereo is what he checks himself against.