1.As-Salaamu Alaikum is a means of immense reward
A man passed by the Messenger of Allaah (Sal Allaahu Alaihi wa Sallam) while he was sitting with others, and said As-salaamu alaykum.
The Prophet (Sal Allaahu Alaihi wa Sallam) said, [He will have] ten hasanaat.
Another man passed by and said As-Salaamu Alaikum wa Rahmatullaah (peace be upon you and the mercy of Allaah).
The Prophet (Sal Allaahu Alaihi wa Sallam) said, [He will have] twenty hasanaat.
Another man passed by and said As-Salaamu Alaikum wa Rahmatullaahi wa Barakaatuh (peace be upon you and the mercy of Allaah and His blessings).
The Prophet (Sal Allaahu Alaihi wa Sallam) said,[He will have] thirty hasanaat.
2.As-Salaamu Alaikum is an aspect of Imaan
Once a man asked the Prophet (Sal Allaahu Alaihi wa Sallam) about which aspect of Islam was best. The Prophet (Sal Allaahu Alaihi wa Sallam) replied: "Feeding the hungry, and saying salaam to those you know and those you don't know." (Bukhaari and Muslim)
3.As-Salaamu Alaikum is a means of entering Jannah
The Prophet (Sal Allaahu Alaihi wa Sallam) said: You will not enter Paradise until you believe, and you will not believe until you love one another. Shall I not tell you about something which, if you do it, you will love one another? Spread salaam (the greeting of peace) among you." (Muslim)
4. As-Salaamu Alaikum gets rid of sins
The Prophet (Sal Allaahu Alaihi wa Sallam) also said, When two Muslims meet (give salaam), and shake hands, they are forgiven their sins before they part (with each other). (Abu Dawud)
5. As-Salaamu Alaikum is the greeting of the people of Jannah
Then, those that feared their Lord shall be driven in companies into Paradise. When they draw near its gates will be opened, and its keepers will say to them: "Salaamun 'Alaikum" (Peace be upon you), you have done well. Enter and live in it for ever. (Surah Zumar:73)
6. Even Allaah (swt) says As-Salaamu Alaikum
Allah sends His Salaam upon the people who believe in Him, especially the prophets such as Nuh, Ibraheem, Ismaeel, Musa, etc.(Surah as-Saaffaat) as well as other pious people.
Jibreel came to house of the Prophet (sal Allahu alayhi wa sallam) and Khadeeja was there. Jibreel said: Allah sends his salaam to Khadeeja. The Prophet (sal Allahu alayhi wa sallam) said: "O Khadeeja, Allah is sending His salaams to you." She said, "Allah is As-Salaam. And upon you O Jibreel, be as-salaam and upon you Rasulullah be as-salaam." [Bukhaari]
It is obligatory to return the greeting of As-Salaamu Alaikum
The Muslim has five rights over his fellow-Muslim: he should return his salaams, visit him when he is sick, attend his funeral, accept his invitation, and pray for mercy for him [say Yar ha'mu kallaah when he sneezes.(Bukhaari, Muslim)
We should say As-Salaamu Alaikum when we enter our homes When you enter houses, greet (with peace) one another with a salutation from Allah, blessed and good. As such Allah makes clear to you His verses so that you understand. (Surah an-Nur: 61)
As-Salaamu Alaikum is also to be said at time of leaving
The Prophet (Sal Allaahu Alaihi wa Sallam) said: "When one of you joins a gathering, he should greet those present; and when he leaves them he should salute them, because the first salutation is not better than the last one." (Abu Dawud and Tirmithi)