Esagerato means "exaggerated" or "over the top." It's a descriptive word that goes beyond common adjectives, making it more interesting and memorable.
In the song, Max Pezzali uses it to describe a love so intense and powerful that it seems "esagerato ed impossibile" (exaggerated and impossible). This word perfectly captures the dramatic and all-consuming nature of the love he's singing about, making it a key descriptor of the song's central theme.
A whole song comes out of a two second glance. Max Pezzali sees an ex in her car, she does not look up, and that is the entire plot: "Ti ho incontrata, ma tu non mi hai visto" (I ran into you, but you didn't see me). He is not even sure what his body did about it. "Ma il mio cuore si è come bloccato / O era fermo prima e ha ripreso a battere" (But my heart kind of froze / Or it had stopped before and started beating again) is a good joke on himself, because he cannot tell whether the shock stopped his heart or restarted it after years of nothing.
Then he oversells the relationship, on purpose. The chorus reaches for the biggest thing available: "Eravamo quel che tutti sognano" (We were what everyone dreams of), the love "che i cantanti cantano" (that the singers sing about), so large "Che potrebbe scomparire l'universo tranne noi" (that the universe could disappear except us). The second half is more honest about what that costs him. He calls his memories "lame" (blades), admits "forse li cerco io" (maybe I'm the one looking for them), and lets his head finish the sentence: "Ciò che è rotto ormai non si riparerà" (what is broken now will never be mended). His heart refuses to hear it. Listen to the last chorus and the tense has quietly moved: "Eravamo" becomes "E saremo quel che tutti sognano" (And we'll be what everyone dreams of). He has talked himself out of the past tense in under four minutes.