God commanded the Jewish people to learn from the moon and renew themselves. In a world that proceeds as though everything is predetermined, we are instructed to sanctify the new month and to believe in constant renewal.

The pagan outlook espouses preserving the status quo, not renewal.

Paganism does not recognize anything new, not on earth, nor in people, nor in the gods and idols it sets above the world and humankind. Everything has iron necessity: Today evolves from yesterday, and tomorrow from today, necessarily. Just as paganism denies the concept of creation ex nihilo, i.e., unrestricted creation in accordance with the free will of the Creator, so too it rejects the notion of creation ex nihilo with regard to one’s moral state, [i.e., the ability to bring about personal change,] and with regard to one’s fate. Guilt and evil are forced to generate guilt and evil, and so forth. Accordingly, a person’s heart lacks any connection to a free, unlimited God; according to it [the pagan worldview], there is no free God ruling in and over the world. Everything is swept down an unseeing, unchanging, and unavoidable course. All freedom is just an illusion; anything new is an extension of the old.

Further reading: For more on Rosh Hodesh and its prayers and customs, see A Concise Guide to Halakha, p. 237.

The mitzva to sanctify the new month was given specifically in Egypt, in order to negate its pagan worldview:

Therefore, it states [in the verse containing the commandment to sanctify the new month] “in the land of Egypt” (Exodus 12:1), i.e., in the land that consistently embraced paganism, in the land where pagan stagnation spread even to the social structure of the region, creating the restrictions of the class system. It was in Egypt, therefore, that God called the future leaders of His nation [Moses and Aaron], and showed them the crescent of the moon struggling to escape from the darkness into a new light, and said: This will be an example for you. Just as this one, which is restrained, [i.e., covered and hidden,] returns to its youth, so shall you take renewal for yourselves, in freedom. Whenever it appears, it will remind you of the renewal of your youth in freedom. When I renew you and when you renew yourselves, you, like this moon, will pass through the night sky of the nations, proclaiming everywhere the teaching of renewal; the teaching of God, who creates in freedom and brings us out into freedom; the teaching of man who receives from the hands of this God the freedom of morality and the freedom of destiny. This renewal is in fact directly derived from the meaning of the name heh-vav-yod [the Tetragrammaton], the God who brings about, through freedom, every moment of the future. Every Jew calls God by this name when he serves Him silently in the depths of his heart. (Rabbi Samson Raphael Hirsch, Exodus 12:1)

Each month constitutes a unique path of renewal, and the Jewish people are commanded to find the opportunity for renewal every month:

These twelve days, the first day of each month, are twelve channels, or gates, for bringing renewal into the world of nature. The Jewish people brought this about, meriting to open up these pathways. With regard to this it is said [in the Musaf prayer on Rosh Hodesh]: “You have chosen Your people Israel from all nations, and have set for them rules for the New Moon.” (Rabbi Yehuda Aryeh Leib Alter, Sefat Emet, Bo)

The moon is renewed each time as it is once again illuminated by the sun. Likewise, the Jewish people are renewed because they again receive the light of God:

“This month [Nisan] is…the beginning of months” (Exodus 12:2), because it is the beginning of all [the moon’s] renewals. For each renewal occurs through receiving light that the sun emits in accordance with its orbit. As the sun emits light in that month, i.e., at that time of the year, with the changes of its course in the different periods of the year, so it is with the renewal of the moon, which receives from [the sun]. Likewise, the renewal of the Jewish people is through receiving the light of the Lord [YHVH], who illuminates them with the emanation of His light, emanating conceptual light upon the lower world. This is the light of His Shekhina, which rests in the souls of the Jewish people, such that they are able to attain this light. (Rabbi Tzadok HaKohen of Lublin, Maĥshavot Ĥarutz 7)