It isn't usual for the Torah to use the term "לחטאת להשם", as it does by Rosh Chodesh. It implies that Hashem needs atonement for something. Reish Lakish explains that the reason it uses this expression regarding the goats of Rosh Chodesh - אמר הקב"ה שעיר זה יהא כפרה עלי שמיעטתי את הירח. The goat of Rosh Chodesh is so to speak a kapara for Hashem for reducing the moon. Tosafos is bothered by the implication that Hashem requires some atonement. Tosafos explains that really the goat is a kapara for the Jewish people, and it is Hashem's responsibility to arrange this atonement. He arranged it specifically on Rosh Chodesh to appease the moon for having to reduce it in size (since originally it was the same size as the soon - chulin 60b).
The Meshech Chochma (Pinchas) explains that the reduction of the moon created a problem in the world. So long as the sun and moon were the same size, it would have been difficult to attribute any power or deity status to either one - there can't be two G-ds. The reduction of the moon enables people to confuse the sun with G-d Himself and start praying to the sun. Had the moon and sun been the same, there would be less idolatry in the world, but now that the sun is larger there is more idolatry and more rejection of Hashem.
I would like to add to the Meshech Chochma. When we say that Hashem needs atonement, it means that Hashem is taking responsibility on some level for aveiros. It is similar to the tefillos of Moshe Rabbeinu using the parable of a king dressing up his son and placing him in front of a beis zo'nos - מה יעשה הבן שלא יחטא. Here too, Hashem is taking responsibility for the aveiros committed due to the reduction of the sun, as if He Himself caused people to go astray and worship the sun. The concept of bringing a korban on the renewal of the month is a method of empowering the moon, trying to restore it's honor. We don't bring korbanos on the renewal of the sun, only on the renewal of the moon - for the purpose of trying to empower the moon and equate it to the sun. But, we recognize the moon by bringing a korban to Hashem to show that just as Hashem controls the moon and the lunar month, he controls the sun and the solar year. By recognizing Hashem's control of the moon, we make an attempt to fix the problem caused by the reduction of the moon which led to sun worship (see maharatz chiyus quoting the Rambam).