The narration in At-Tirmidhi is: The Messenger of Allah (sallallaahu ’alayhi wa sallam) was asked: "O Messenger of Allah! When two persons meet, who should greet the other first?" The Messenger of Allah (sallallaahu ’alayhi wa sallam) said, "The person nearest to Allah i.e., one who is more obedient and therefore closer to Allah will say: As-Salam first."
Commentary: The degree of a man's humbleness and modesty will be measured by the degree of his nearness to Allah. One who is nearer to Allah is always the first to offer As-Salam to others, while others stick to their stuck-up behavior.