On the words עגל את וישחט, the rishonim explain that Aharon slaughtered the yetzer hara who wanted him to dwell on the golden calf. He said, "Even so, I will go and do Hashem's will."
We say in Haggadah shel Pesach, לעבדתו המקום קרבנו ועכשיו אבותינו היו זרה עבודה עובדי מתחילה. The Machnovka Rebbe zt'l said that his father would say, מתחילה, when a person focuses on the past, what used to be, that is avodah zarah. ועכשיו, but to focus on the present, לעבדתו המקום קרבנו, that will draw a person to avodas Hashem.
The Ramban writes (quoting a Sifra), "Aharon saw the form of an ox on the mizbeiach [which represents the egel] and he was frightened. [He understood that the 'ox' was a sign that he wasn't worthy to bring the korbanos, since he had an association with the formation of the egel.] Moshe came to him and said, 'Aharon, my brother, don't be afraid. Be proud and approach the mizbeiach.'"
The Baal HaTurim teaches that Aharon looked at the קרנות, the corners of the mizbeiach, and they appeared to him like the horns of a calf. This was to break his spirit, to make him feel unworthy of bringing the korbanos. Moshe told him, "Don't look at the קרנות. Go to the mizbeiach. You were chosen to do this service."
The lesson is a reminder for all Yidden that we are worthy to serve Hashem. Hashem chose us to do His service. It is just the yetzer hara who tells us that we aren't worthy of greatness.
Imrei Chanoch
(written by a student of the Avnei Nezer) writes that Chazal (Sotah 5a) tell us that a talmid chacham should have an 1/8th of an 1/8th of pride (שבשמינית שמיני). The words שמיני שבשמינית can be translated as the eighth pasuk of parashas שמיני. This pasuk is "Aharon approached the mizbeiach and slaughtered the egel of his chatas." The lesson is that a person must always lift his moral, and regardless of what occurred in the past, feel confident that he can now serve Hashem properly and well and accomplish great things.
Teshuvah through Love
It states (9:2) "Hashem said to Aharon, 'Take for yourself a calf as a chatas.'" The meforshim explain that the egel was to atone for the חטא העגל. The question is, there is a rule: קטיגר אין סניגר נעשה, something used to prosecute, and remind heaven of our sins, can't be used as an advocate, to remove our sins. How then can the calf rectify the sin of the golden calf?
The Kli Chemdah answers, quoting the Sfas Emes, that Aharon and Yidden did teshuvah with love, and the Gemara (Yoma 86b) states that when Yidden do teshuvah through love, the aveiros become mitzvos. Therefore, an egel is no longer a prosecutor; rather, it is our advocate. Due to their teshuvah, the egel became like a mitzvah.
It states (Yeshayahu 40:2) עוונה נרצה כי, "Her sin has been appeased." The Kli Yakar explains that the sin itself goes up before Hashem and becomes our advocate. So great is teshuvah from love, that the aveiros turn around and become merits. A hint to this concept is the חלבנה, one of the components of the ketores in the Beis HaMikdash. The חלבנה has a foul scent, representing the sinners of Yisrael. But they, too, become part of the ketores, a beautiful scent before Hashem. This is because when baalei aveirah do teshuvah with love, they create a nachas ruach before Hashem.
Moshe told the nation to bring the korbanos he listed because (9:4) "for today Hashem is appearing to you." The Alshich Hakadosh notes that Moshe didn't say "for today Hashem is appearing to us." It seems that Hashem will come to Aharon and to the nation, but not to Moshe. How can that be?
תבואה פרי אך (written by Rebbe Hershele Liske zy'a) answers that Chazal teach us, "The place where baalei teshuvah stand, the greatest tzaddikim can't stand." The Shechinah will come down to dwell with Aharon and the Jewish nation because they did teshuvah. Moshe never sinned with the egel, so he didn't get this privilege.
Moshe said to Aharon (9:7) "Approach the mizbeiach" to bring the korbanos. Rashi explains that Aharon needed encouragement: "Aharon was embarrassed and he was afraid to approach the mizbeiach [due to his association with the egel]. Moshe told him, 'Why are you embarrassed? You were chosen for this!'"
The Chasam Sofer explains that Moshe was telling Aharon that his merits are greater than his, because Aharon had the merit of teshuvah with love. Therefore, this aveirah became a merit. So why are you embarrassed? This matter, which you think is to your detriment, is the reason you were chosen.
The Alshich HaKadosh (Vayikra 7:2) advises people to do teshuvah with love before they die, because then they will benefit even from their aveiros in Olam HaBa. We quote: "One should do teshuvah from love before he dies, and then it will turn over and become his merits, and he will eat and enjoy the aveiros themselves. Instead of being an impure sin, it is now kodesh kadoshim, because it turned over to be holiness and merit."
It states (9:24) "A fire went forth from before Hashem and consumed the olah and the fats upon the mizbeiach, and all the people saw, sang praises, and fell upon their faces." The Divrei Shmuel writes in the name of the Alshich HaKadosh that "they fell upon their faces" means that they fell on their faces and did teshuvah.
In this episode, there were two times that the nation wanted to do teshuvah. The first time was when the fire didn't initially come down onto the mizbeiach (see Rashi 9:23). The nation understood this was a sign that Hashem wasn't happy with them and that He wouldn't dwell among them. The nation wanted to fall on their faces and do teshuvah, but they held themselves back. This is because they were so broken at that time, they knew that their teshuvah would lead them to sadness and depression.
After the fire miraculously descended from heaven on the mizbeiach, they wanted to do teshuvah a second time. This time they wanted to do teshuvah for their initial thoughts and fears, for having thought that Hashem wouldn’t reside among them. They were embarrassed that they thought so. And that time, they succeeded in doing teshuvah, because it was teshuvah amid immense joy and singing, as it states "They sang praises." Now they could fall on their faces and do teshuvah because the teshuvah won't result in despair, chalilah.