T'esquinter is a reflexive verb meaning 'to wear oneself out' or 'to damage oneself'. It's a less common, more evocative word than simply 'vieillir' (to age).
In the song, Slimane sings, "Voir le temps t'esquinter, comme il le fera pour moi" (To see time wear you out, as it will for me). He uses this word to express a deep, shared experience of aging and the physical toll time takes, emphasizing a desire to witness and share every aspect of life with his loved one, even its hardships.
Slimane's whole song is a wish list for growing old, and he says the wish plainly, over and over: "Je veux être vieux" ["I want to be old"]. Most songs about aging are really about being afraid of it. This one isn't. He wants the proof of it on his body, "Et avoir des rides plein l'visage" ["And have wrinkles all over my face"], "Des rides sur les mains" ["Wrinkles on my hands"], lines that "trahissent mon âge" ["betray my age"]. A wrinkle here isn't something to fix. It's a receipt for time spent living.
What he actually wants old age for is other people. He wants to watch someone he loves grow old too, "Voir le temps t'esquinter" ["Watch time wear you down"], the same way it will eventually happen to him. He wants to meet "les enfants de mes enfants" ["my children's children"], and further out still, "les enfants d'leurs enfants" ["their children's children"]. And underneath all of it is one blunt reason: he wants to be old "Pour défier le diable et le temps" ["To defy the devil and time"]. Surviving, here, is the whole victory. By the end he's only asking to have "fait d'mon mieux" ["done my best"], and the song cuts off mid-sentence on "Je veux être" ["I want to be"], like finishing the thought out loud would jinx it.