It's just a new month. Why does a date on the calendar get its own prayer?
Start with the first commandment the Jewish people ever received. It is about the calendar. In Egypt, just before the Exodus, God says: "This month is for you, the first of the months" (Shemot 12:2). Ramban (Rabbi Moshe ben Nachman, a Spanish Torah commentator who died in 1270) says plainly that this is the first commandment God gave Israel.
Rashi (the great French commentator on the Torah, who died in 1105) opens his commentary on the Torah with the same point. He quotes Rabbi Yitzchak, an early Sage, who said the Torah could have started right there, at "This month," since that is the first commandment given to Israel.
Why would a calendar come first? Sforno (Rabbi Ovadiah Sforno, an Italian Torah commentator of the 1500s) answers: from now on the months are yours, to do with as you wish. In the days of slavery, your days were not yours. They belonged to the work and the will of other people.
Ramban adds that the Torah gives the months no names. It only counts: the first month, the second month, counting from the Exodus. So every time a month was mentioned, the miracle was remembered. The names we use today came back with the Jews from Babylon. The Yerushalmi (the Gemara of the Land of Israel) says so, and Ramban explains why: so that we remember where God brought us up from.
The prayer then asks for what fills those days: joy, gladness, a living, peace, forgiveness.
Shemot 12:2,Rashi on Bereishit 1:1,Sforno on Shemot 12:2,Ramban on Shemot 12:2,Yerushalmi, Rosh Hashanah 1:2The blessing says people make the months holy. Isn't that God's job?
The prayer ends with a blessing: "who makes Israel holy, and the new moons." The Gemara notices a problem. A blessing should end with one theme, not two (Berakhot 49a). Yet Rav, a leading Sage in Babylonia, ended the Rosh Chodesh blessing this way (Berakhot 44a).
The Gemara answers in a few words: Israel are the ones who make the new moons holy. So it is one theme. Rashi explains that the festivals depend on the court, which declares each month holy after witnesses see the moon. Shabbat is different. It is holy on its own.
The Mishnah (the first written collection of Jewish oral law, about 1,800 years old) describes it. The head of the court says "It is made holy," and all the people answer, "It is made holy, it is made holy." One Sage, Rabbi Elazar son of Rabbi Tzadok, says that if the moon was not seen on time, the court does not declare it, because Heaven has already made it holy.
The hardest test came when Rabbi Yehoshua, a leading Sage, thought the court had accepted the wrong witnesses. Rabban Gamliel, the head of the high court, ordered him to show up, with his staff and his money, on the day that was Yom Kippur by Rabbi Yehoshua's own count. To him that meant breaking Yom Kippur, and he was distressed.
Rabbi Akiva, another leading Sage, comforted him with a verse about the festivals: "which you shall proclaim them" (Vayikra 23:4). In the Gemara's telling, he pointed out that the Hebrew word for "them" is spelled the same as the word for "you." Read it as "you," he said: you, even by mistake; you, even on purpose; you, even if misled by false witnesses.
Rabbi Yehoshua went. Rabban Gamliel stood up and kissed him on the head: "Come in peace, my teacher and my student." On this reading, the blessing keeps both halves. It is addressed to God, who makes Israel holy. And Israel make the months holy.
Berakhot 44a and 49a,Rashi on Berakhot 49a,Mishnah Rosh Hashanah 2:7 and 2:9,Rosh Hashanah 25a,Vayikra 23:4The month just started. What am I being forgiven for?
Look at the end of the list. After joy, rescue and a living, the prayer asks for forgiveness twice, three times in a leap year. But the month has only just begun. What is there to forgive? The Musaf prayer for this same day gives a first answer: "You gave Your people the new moons, a time of atonement for all their offspring."
In the Temple, a goat was brought as a sin offering every Rosh Chodesh. Abudarham says it atoned for the Temple and its holy things becoming impure. In the Gemara, Shmuel (a leading Sage in Babylonia) explains the view of Rabbi Yehuda (a Sage of the Mishnah) from the words "to Hashem": a wrong only Hashem knows about. The case there is narrow: someone impure entered the Temple or ate holy food, and never found out.
Abudarham adds a second view. Some explain that the Rosh Chodesh burnt offering atoned for "what the days of the month give rise to."
Rabbi Yehudah HaLevi (a Spanish poet and philosopher of the 1100s) reads it this way in the Kuzari, his book explaining Jewish belief. He lays out a rhythm of repair: Shabbat for the week, Rosh Chodesh for the month, then the festivals, and finally Yom Kippur. For Rosh Chodesh, he says, "their offspring" means the offspring of the months, the new things the days bring, and he quotes Mishlei: "you do not know what a day will bring forth" (27:1).
A custom grew from this. Some people fast the day before Rosh Chodesh and say prayers for forgiveness, called Yom Kippur Katan, the "small Yom Kippur." The Kitzur Shulchan Aruch (a short guide to daily Jewish law from the 1860s) says the month's wrongs are forgiven that day, like the Rosh Chodesh goat, and that each place follows its own custom.
So these sources do not say you did something terrible last month. They describe Rosh Chodesh as a fixed time for clearing away what the weeks left behind, including what nobody noticed and what you could not have planned.
Siddur Sefard, Rosh Chodesh Musaf,Abudarham, Rosh Chodesh 14,Shevuot 9a,Kuzari 3:5,Mishlei 27:1,Kitzur Shulchan Aruch 97:1