Kiddush Levana or Blessing of the Moon (Birkat HaLevana) is seemingly small rabbinical mitzvah based on Talmud Sanhedrin:
Rabbi Yohanan is quoted in the Gemara Sanhedrin 42a saying: “One who makes the blessing for the new moon in its proper time is considered as if he welcomed the Shechinah [the Divine Presence]”
The parallel is drawn between the physical moon on one side and Shechinah/Jewish people on the other.
Just like the physical moon is constantly going through the cycle of visible waxing and waning, so does Klal Isroel/Shechina is always going through ups and downs, reflecting the G-d’s light more or less, channeling more Godliness into the world or less.
The more accurate translation of the above-mentioned verse from Gemara Sanhedrin, which does give a hint of the deeper meaning of the mitzvah Kiddush Levana, is:
“One who makes the blessing for the new moon in its proper time is considered as if he received/accepted the Shechinah”.
So rather than ambiguous “greeting”, we are talking about Kiddush Levana being a procedure for making a connection with the Shechinah!
When the Kiddush Levana is done at the right time and in the state of joy, the connection with the Shechinah allows us to give it strength on one side, and allows Shechinah to affect our day-to-day life, filling in our material and spiritual lack. In other words, we benefit from boosted Hashgacha Pratit – individual Divine Providence in our life.
So “receiving” Shechinah as a consequence of Kiddush Levana is quite literal – one receives Shechinah into his life in the month to follow.
The “greeting” translation is quite accurate as well, as “greeting” implies exchange of energy, which is what is happening during Kiddush Levana.
We can glean additional meaning of the mitzvah if we look closer into the specifics of time when Kiddush Levana is done: from the 7-th day to the 14.5 days of the lunar cycle. This means we only “make the contact” when moon is growing but already reached certain maturity.
This timing allows us to “hitch the hike” of the growth/expansion energy radiated by the moon/Shechinah.
(It also explains why, according to Halacha, one cannot make Kiddush Levana after the moon starts waning – instead of the Chesed/Expansion momentum, after the 14.75 days of the moon cycle, one is bound to get Gevurah/Contraction energy. Some smartphone weather apps show moon phases along with exact time of the cycle, for example “weather & clock widget” for Android. If making your Kiddush Levana on 15-th day of the lunar month, you should check that 14 days and 18 hours mark has not yet passed)
So this brief prayer service becomes very important factor when one is trying to align one’s life with the moadim and zmanim (annual, monthly and weekly time cycles), to ensure that we are riding the wave, not crashing into it.
See also the related topic on Rosh Chodesh.
Halachic details along with recommended Kavanot (intentions) – “Ben Ish Chai” on the Kiddush Levana:
Good in-depth article on Kiddush Levana addressing other aspects of the mitzvah.