Kommentar zu Wajikra 16:15

אשר לעם [THE GOAT OF THE SIN-OFFERING] THAT IS FOR THE PEOPLE — Only for that which the bullock effects atonement for the priests (i. e. for ומאת מקדש וקדשיו‎ט) the goat effects atonement for the Israelites. It (the goat spoken of here) is that goat upon which the lot “for the Lord" fell (Yoma 61a; Shevuot 2b).

החטאת אשר לעם. After he had completed offering the sin offering for his own atonement, also the people would be on the level where they could face the “King.” The sequence described here which at fist glance might seem as if the High Priest were more concerned with his own atonement than that of the people whom he represented, is confirmed by the prophet (Tzefaniah 2,1) in the words התקוששו וקושו, which is interpreted in Sanhedrin 18 as “attend to your own integrity before you attend to the integrity of others.”

The goat atones. I.e., just as the bullock atones for impurity of the Temple and its holy things for the kohanim as Rashi just explained above, similarly the he-goat of the public atones for the impurity of the Temple. Rashi is [actually] saying [the reverse]: You might ask: Why does the verse write “that belongs to the people” since there is no he-goat sin-offering that does not belong to the people? This comes to teach you, what does the bullock atone for on behalf of the kohanim? That same thing which the he-goat of the people atones for. The he-goat of the people certainly atones for the impurity of the Temple as it is written regarding it, “He will atone for the Sanctuary, from the impurities of Bnei Yisroel,” so the bullock of the kohanim atones for the impurity of the Temple. [See Re’m]

אשר לעם, “intended for atonement of the people,” and not for other ordinary priests, not of the High Priest’s family. The question then arises by what means did those priests receive their atonement? Answer: by the bull that brought atonement of Aaron’s sins.

כאשר עשה לדם הפר AS HE DID WITH THE BLOOD OF THE BULLOCK, viz., sprinkling once upwards and seven times downwards (Yoma 53b).

One above and seven below. Explanation: Rashi needs to explain this so that we do not learn the first instance of “blood” [in verse 14] from the last instance [of “blood” in verse 14] and the last instance [of “blood” in verse 15] from the first instance [of “blood” in verse 15], and say that the above [sprinkling] and below [sprinkling] of both the bullock and the he-goat] should be the same, or one [sprinkling above and below for the he-goat] or seven [sprinklings above and below for the bullock]. We would learn each case from itself. Therefore “just as he did with the bullock’s blood” tells us that we compare the he-goat to the bullock in all its actions, [and both of their numbers of sprinklings are equal]. [See Re’m verse 14]