Hachodesh Hazeh Lachem – Moreinu HaRav shlit”a
החדש הזה לכם ראש חדשים ראשון הוא לכם לחדשי השנה,
This month shall be for you the beginning of the months,
it shall be for you the first of the months of the year.
The Medrash states,
R’ Levi in the name of R’ Yossi bar Ilai said ‘it is appropriate that the bigger one count according to the bigger and the smaller one to count according to the smaller. Esav counts according to the sun which is bigger, and Yaakov counts according to the moon which is smaller.’ R’ Nachman said, ‘this is a good sign, that Esav follows the sun [solar calendar] which is great. For just like the sun rules at day but not at night, so too Esav has a portion in This World, but none in the World to Come. [Whereas] Yaakov follows the moon [lunar calendar] which is small. Just like the moon rules at night but also at day, so too Yaakov has a portion in This World and in the World to Come.’ R’ Nachman said further ‘as long as the light of the bigger one exists, the light of the smaller one isn’t visible. Once the light of the bigger one fades, the light of the smaller one becomes visible. So long as the light of Esav exists, Yaakov’s light is not visible. But once Esav’s light fades, Yaakov’s light becomes visible, as the passuk states,קומי אורי כי בא אורך…כי הנה החושך יכסה ארץ וערפל לאומים ועליך יזרח ה’ וכבודו עליך יראה “Arise! Shine! For your light has arrived… For behold! Darkness may cover the earth and a thick cloud [may cover] the kingdoms, but upon you Hashem will shine, and His glory will be seen upon you.”
Says the Rama:
It is common practice to say [at Kiddush Levana] דוד מלך ישראל חי וקיים, “David king of Israel is alive and in existence,” because the kingship of [David] is likened to the moon and will be renewed in the future [like the moon is renewed]. The community of Israel will once again cleave to her husband, the Holy One Blessed is He, just like the moon renews [its relationship] with the sun, as the passuk states, שמש ומגן ה’, “a sun and a shield is the Lord”. This is why we rejoice and dance at the sanctification of the new moon, which resembles the rejoicing at a wedding.
The moon is substantially smaller than the sun, and even so, the nations of the world are likened to the sun whereas the Jewish people are likened to the moon. This is because the moon has a unique advantage over the sun in that she alone rules at night (whereas during the day, the sun’s domain, the moon is often visible). Just like the sun, the nations of the world are great and numerous, but they rule only at day. Every nation has its heyday, so to speak, every empire has its zenith, but times change; their sun sets and they fall, sinking to the depths of oblivion and obscurity. They are relegated to the realm of the insignificant and soon forgotten. But Am Yisrael, the smallest of the nations, like the moon, rules at night as well. All through the long night of exile, through the terrible darkness of G-d hiding His face, Am Yisrael is still in existence, awaiting dawn. נפשי לה’ משומרים לבקרI yearn for my Lord among those longing for the dawn, and that dawn will come, “שהם עתידים להתחדש כמותה ולפאר ליוצרם על שם כבוד מלכותו” those who are destined to renew themselves like it [the moon] and to glorify their Molder for the name of His glorious kingdom.
Rav Yaakov Emdin writes
I swear! When I look at our nation’s existence throughout the exile, one sheep among seventy wolves, this miracle is greater in my eyes than all the miracles performed for our forefathers in Egypt, at the [Red] Sea, and in Eretz Yisrael.
This gift which G-d has bestowed on the Jewish people—that we are likened to the moon which rules at night—represents the greatest miracle of all miracles. As He has promised, אז יבקע כשחר אורך וארוכתך מהרה תצמח then your light will burst out like the dawn and your healing will speedily sprout. This is why we say at Kiddush Levana, דוד מלך ישראל חי וקים David, king of Israel is alive and enduring. As the Rama writes that David’s kingship is also compared to the moon. Even though kingship has ceased from Bnei Yisrael and their temple has become desolate, nevertheless we await this future renewal, and even though He may tarry, nevertheless we anticipate every day that He will come and Moshiach ben Dovid will renew his kingship speedily.
In future times, the moon, too, will be renewed and once again G-d will make it large like it was initially at creation. But this is contingent on us and our actions. When we return to G-d and fix our ways and deeds, the moon too will return to its original glory.
Our Sages tell us,
Reish Lakish said: “why is the he-goat of Rosh Chodesh different from other mussaf he-goats in that it is stated regarding it ‘(לחטאת) לה, )atonement( for G-d? For by this expression, the Holy One Blessed is He, said “This he-goat shall be an atonement for that which I diminished the size of the moon.
The K’sav Sofer asks the following in the name of his father the Chassam Sofer: why is the Rosh Chodesh atonement incumbent on Bnei Yisrael? It would seem to be G-d’s responsibility, for having diminished the moon. He explains that were we to fix ourselves and our deeds, the moon would return to its original state, shining bright as the sun, and because it is due to our shortcomings and inadequacies, we are responsible to bring this atonement for the fact that the moon is still diminished. 
Rav Isser Zalman Meltzer says it is customary that one look at the tzitzis fringes after Kiddush Levana. He says that although this may seem strange since nighttime is not the time to don or look at tzitzis [as it says וראיתם אותו וזכרתם, you will see it (tzitzis) and remember, and at night one cannot adequately see], in the future, when the moon will return to its former glory, and nighttime will be as bright as day, then nighttime will also be a time to don tzitzis. This is why we look at the tzitzis fringes after Kiddush Levana, hoping, planning, and praying for the day when we will be able to observe the mitzvah of tzitzis even at night, for night will be as bright as day.
Those who are destined to renew themselves like it [the moon] and to glorify their Molder for the Name of His glorious kingdom.
ששים ושמחים לעשות רצון קונם
They are joyous and glad to perform the will of their Creator
Bnei Yisrael are compared to the moon. Though one may wonder why they’re compared to the moon and not to the sun which is much larger, there is a profound reason for this. The sun is large, but it does not grow; it is static, whereas the moon is not—it wanes and waxes. What this represents is that it is not merely a question of how big a person is, but whether a person is a growing person.
The Maharsha writes that although G-d diminished the moon, the moon is happy and joyous to do His will. Our Sages tell us that G-d said to the moon “go and diminish yourself.” This is a most difficult test; G-d did not diminish the moon Himself rather commanded her to diminish herself. Regarding this characteristic we say at Kidddush Levana ששים ושמחים לעשות רצון קונם, they are joyous and glad to perform the will of their Creator. We learn an important lesson from the moon. At times one must diminish or negate oneself, and do so happily.
An explanation is offered in the name of the Holy Rebbe Menachem Mendel of Rimnov regarding the Mishna ומורא רבך כמורא שמים, and the reverence for your teachers [should be] like the reverence of Heaven. He explains that we must learn from שמים, the heavenly bodies, the sun and the moon, how a student must behave towards his teacher. The moon does not have its own light—it merely reflects the light of the sun. It is truly a wonder that as the moon gets closest to the sun (when it is positioned between the earth and the sun), it does not reflect any light, whereas when it is furthest from the sun, when it is beyond Earth, it is a full moon shining brightly. So, too, must a student sometimes stand from a distance, not get too close to the point where he is cavalier with his teacher, and only then, from a respectful distance, can he attain the wealth his teacher has to offer. The reverence for your teachers [should be] like the reverence of Heaven.
שהם עתידים להתחדש כמותה ולפאר ליוצרם על שם כבוד מלכותו,
Those who are destined to renew themselves like it,
and to glorify their Molder for the name of His glorious kingdom.
 קידוש לבנה
 שמות רבה ו’:ג’
 ישעיהו ס’: א’-ב’
 רמ”א או”ח תכ”ו:ב’
 קידוש לבנה
 סולם בית אל
 ישעיהו נ”ח:ח’
 שבועות ט’ ע”א
 שו”ת כתב סופר, או”ח סימן ע”ד
 The printers of the Vilna Shas were wise to comment on the side of this Gemara in שבועות ט’ ע”א, that this Gemara has deeper kabalistic meanings to it, and cannot be taken at face value to imply that G-d needs an atonement, ח”ו.
 Brought down in הר צבי ח”א סימן י”ב
 קדוש לבנה
 סנהדרין מ”ב ע”א
 חולין ס’ ע”ב
 פרקי אבות ד’:ט”ו
 קדוש לבנה