When you meet each other, do not shout hello or bye-bye, or some silly chatter.
Let the moment of meeting be sanctified by the remembrance of God.
Say, Ram Ram, or Om, or Hari Om, or Sai Ram.
Sathya Sai Speaks X, Chap. 14
The quote from Swami (above) states in a nutshell why we greet someone with "Sai Ram". It is a remembrance of God, the one God that IS the other person and ourselves.
In Bavaria, southern Germany, many people greet each other with "Gruess Gott", which means "Greet God". Same idea!
When greeting someone with "Sai Ram", try to think of greeting not only the worldly body that you see but our SAI in that person. We could use any name of God; it is the intent and feeling that counts. We have happened to choose "Sai Ram" because of the universal appeal of that name as a mantra. (Some devotees do repetition of the Name --namasmarana-- with "Om Sri Sai Ram", repeating "Om Sri" on the inbreath and "Sai Ram" on the outbreath.)
The word "Sai" can be split into two: "Sa" and "Ai". We have been told that "Sa" means universal father, while "Ai" stands for universal mother. Sathya Sai Baba, our Sai, is our mother and father.
"Ram" can be viewed as a shortening of "Rama". One interpretation of this word is that "Ra" is the Fire Principle, which burns all to ash, while "ma" stands for maya or illusion, so together, they mean the destruction of illusion.
Sai Baba has said also that "Rama" means "He who pleases, fills with bliss, causes delight", that "Rama" means the joy that comes from love.