Hadith: The closest people to Allah are the ones who give the Salām first

عن أبي أمامة رضي الله عنه قال: قال رسول الله صلى الله عليه وسلم : «إن أَوْلَى الناس بالله من بَدَأَهُمْ بالسلام». وفي رواية للترمذي: قيل: يا رسول الله، الرَّجُلان يَلْتَقِيَان أَيُّهُمَا يَبْدَأُ بالسلام؟، قال: «أَوْلاهُمَا بالله تعالى».
[صحيح] - [الرواية الأولى رواها أبو داود. الرواية الثانية رواها الترمذي]

Abu Umāmah (may Allah be pleased with him) reported that the Prophet (may Allah's peace and blessings be upon him) said: "The closest people to Allah are the ones who give the Salām first" [Abu Dāwud]. In the narration of At-Tirmidhi: The Messenger of Allah (may Allah's peace and blessings be upon him) was asked: "When two persons meet, who should give the Salām first?" He said: "The one who is closer to Allah."
[Authentic hadith] - [Narrated by At-Termedhy - Narrated by Abu Daoud]

Explanation

The best of people and those who are the most obedient to Allah are the ones who hasten to greet their brothers with Salām (saying: As-Salāmu ‘alaykum). This is because he hastens to do an act of obedience to Allah, seeking Allah's reward. This is what makes him closest to Allah and most obedient to Him.