There are a whole bunch of Pesukim that we say 3 times at Kiddush Levana.
1 - Why do we say these specific Pesukim 3 times?
2 - Specifically I am curious why we say Shalom Aleichem 3 times?
The Perisha (Orach Chaim 426) says that we say Shalom Aleichem three times is because we previously cursed our enemies with "Tippol Aleihem." Thus we are assuring our friends that we wish peace for them.
However the Perisha does not explain why the other verses are said three times, including "Tippol Aleihem," which is the reason for saying "Shalom Aleichem" three times.