“saw how he died.” The Greek is literally, “breathed out,” which is a euphemism, in which “breathed out” was used to mean “die.”a The centurion had been there watching Jesus and saw the way in which he died. He heard the things he said and saw what happened as he was on the cross. From that, he concluded that what he certainly must have heard about Jesus was true, that he was the Son of God.
“Truly this man was the Son of God.” The claims of Jesus to be the Son of God would have been well known, as well as the miracles he did, and the fact that the religious leaders wanted him crucified because they envied him. Thus it is not hard to believe that the centurion, upon seeing the love of the Christ (Father, forgive them, etc.), his bravery, and all the miracles and signs that accompanied his death, would be convinced that this man was in fact who he claimed to be, and indeed, who the sign over his head said he was.