Frodo saw what no one else did.
When Boromir followed him, he saw a monster beneath the man’s face, saw the lust for the Ring in all its ugliness. He knew in that moment what Bilbo had felt when he saw the Ring in Frodo’s possession. He knew what it did to all who touched it.
He saw the beast as Boromir talked to him, asking him questions as a distraction while his eyes stayed on the Ring. He saw the evil.
He put his hand on the chain and let it slide down to touch the Ring. The Precious.