Stringere means 'to hold tight' or 'to clasp'. While it can be used in everyday contexts, its specific use in this song is particularly evocative and intimate.
In Gino Paoli's
Gino Paoli spends this whole song answering one question: what is any of this for? He opens by admitting he had no idea. 'Io non sapevo per cosa vivevo' [I didn't know what I was living for], he sings, then works through the answer body part by body part. His eyes are for seeing her. His lips are for speaking to her. His ears are for hearing her voice. And his hands, 'Le mie mani mi servono' [My hands serve me], are for holding her. The lines barely change from one to the next; that's exactly the point. He's checking every part of himself and getting the same answer every time.
The chorus swaps one word for another, and that's the whole turn of the song. First he calls himself 'Un uomo vero perché ti amo' [A true man because I love you]. A few lines later he trades it for 'Un uomo vivo perché ti amo' [A living man because I love you]: loving her makes him real, then it makes him alive. Somewhere past the two-minute mark the song stops introducing new lines at all. It just repeats 'Perché ti amo, io ti amo' [Because I love you, I love you] and 'Io ti amo e tu ami me' [I love you and you love me] until the music runs out, like he already said everything else and this is what's left.