Why say As-Salam?

السلام عليكم
بِسْمِ اللهِ الرَّحْمنِ الرَّحِيمِ
"But when you enter the houses, greet one another with a greeting from Allah (i.e Asalamu Alaikum) blessed and good." (24:61)

"When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally." (4:86)

Abdullah bin Salam reported: I heard the Messenger of Allah (صلى الله عليه و سلم ) saying,"O people, exchange greetings of peace (i.e, say: As-Salamu Alaikum to one another), feed people, strengthen the ties of kinship, and be in prayer when others are asleep, you will enter Jannah in peace," [At-Tirmidhi]

Hadith # 849 Page # 701 Riyad-Us-Saliheen.

Ana bin Malik reported: Messenger of Allah (صلى الله عليه و سلم ) said to me, "Dear son, when you enter your house, say As-Salamu Alaikum to your family, for it will be a blessing both to you and your family." [At-Tirmidhi]

Hadith # 861 Page # 709 Riyad-Us-Saliheen.

Abu Umamah Sudaiy bin Ajlan Al-Bahili reported: The Messenger of Allah (صلى الله عليه و سلم ) said, "The person nearest to Allah is one who is first to offer greeting." [Abu Dawoud]

The narration in At-Tirmidhi is: The Messenger of Allah (صلى الله عليه و سلم ) was asked: O Messenger of Allah! When two persons meet, who should greet the other first?" The Messenger of Allah (صلى الله عليه و سلم ) said, "The person nearest to Allah (i.e., one who is more obedient and therefore closer to Allah will say: As-Salam first."