Vayikra 9:22 records that towards the end of the ceremony on the eighth day, Aharon blessed the people. After Aharon blessed the people, the end of 9:22 records that he finished the sacrifices of the eighth day. According to 9:6, once the sacrifices were finished, then the glory of G-d was to appear. However, the beginning of 9:23 does not record the appearance of the glory of G-d, but that Moshe and Aharon went into the ohel moed, which is usually understood to be the mishkan proper, but I think also means the courtyard of the mishkan, see our discussion above on 1:1, “The terms mishkan and ohel moed in the book of Vayikra.” The remainder of 9:23 records that Moshe and Aharon came out from where they were, and blessed the people and then the glory of G-d appeared to the people.
Various suggestions have been made as to what were these blessings (see Rashi and the Ramban on 9:22,23), but a more basic question is why were the blessings said altogether? 9:1-7 records Moshe’s instructions to Aharon, his sons and the elders for what they were to do at the ceremony of the eighth day, and Moshe did not mention that any blessing was supposed to be said. Furthermore, why did Moshe and Aharon go into the ohel moed? And, why did they bless the people a second time after Aharon had already blessed the people?
Rashi (on 9:23) records two suggestions for why Moshe and Aharon entered the mishkan, as he follows the common approach that the ohel moed and the mishkan are synonyms. One, Moshe wanted to teach Aharon how to offer the spice offering (ketoret), but this is difficult since why at this point would this be important? The glory of G-d was supposed to appear and Moshe was taking time out to teach a lesson concerning the ketoret? It was not recorded in the Torah that the spice offering was to be offered as part of the ceremony of the eighth day, and after the ceremony of the eighth day, Moshe could have taught this lesson. Rashi's second answer (see also Ibn Ezra and Hizkuni on 9:23) is that Moshe and Aharon went inside the mishkan to pray for the appearance of the glory of the G-d. Yet, why was there a need for any prayer given that Moshe had already stated that the glory of G-d would appear due to the bringing of the sacrifices (9:6)? Furthermore, why could they not have prayed outside the mishkan, as the events of the day were focused on the outer altar? It seems that Moshe and Aharon’s entrance into the ohel moed relates to the two blessings since one blessing occurred before they entered the ohel moed and one occurred after they left the mishkan/ ohel moed, but what could be the connection?
In order to understand 9:23, we need to understand why Aharon blessed the people in 9:22. It seems that Aharon decided on his own to state the blessings, as it is not recorded that Moshe had told him to bless the people. While many people would think that giving a blessing is a nice idea, this action was problematic since when one worships G-d one must do what one is commanded and not add extras. A few minutes later this would be the sin of his sons, Nadav and Avihu, who attempted to serve G-d in a way that was not commanded and they were killed for this sin, 10:1,2, see our discussion below on 10:1,2, "Nadav and Avihu" https://lobashamayim.blogspot.com/2010/04/vayikra-1012-shemini-nadav-and-avihu.html.
If Aharon's blessing was a mistake, then Moshe and Aharon went into the mishkan or in the courtyard of the ohel moed to talk. When they were alone, Moshe told Aharon that he is only to do what is commanded and not to add on extras. This rebuke might explain why the Torah did not mention what was said in the ohel moed.
The second blessing when they left the area where they had their brief discussion was an attempt to rectify Aharon's mistake. Aharon had blessed the people even though he had not been commanded to do so, but when Moshe and Aharon blessed the people, the people would think that they had been commanded by G-d when they conversed alone, and then the people would also think that Aharon had acted appropriately when he had had blessed the people in 9:22. (Maybe G-d spoke to them in the ohel moed but this cannot be known since it is not recorded.) The important idea was that the people would learn that one can only do what commanded, even though in this case the people were possibly being fooled. After this rectification of Aharon's mistake, then the glory of G-d appeared in the end of 9:23.
A possible proof for this approach is that immediately prior to Aharon's first blessing, in the end of 9:21, the Torah records that Aharon did exactly as he was commanded by Moshe. (This idea to do as commanded also appears in 9:5,6,7,10.) The statement in 9:21 is referring to Aharon's work with the sacrifices, and the implication is that while this work was commanded, Aharon's blessing that followed this statement was not commanded by G-d or Moshe.
Yet, if Aharon acted inappropriately when he first blessed the people, why was he not punished immediately like his sons were? Maybe his sin was not grievous enough to warrant his death. Or, maybe had he been killed, the ceremony would have stopped without the appearance of G-d's glory. This could not happen after Moshe had announced that G-d's glory would appear, 9:6. With this understanding, G-d delayed the appearance of the glory of G-d until Moshe was able to rectify Aharon's mistake, and then maybe Aharon was not punished later since he already suffered greatly when his sons were killed, though they died for their own sins.
Independent of the question of whether Aharon should have been punished for blessing the people on his own, Aharon, Nadav and Avihu all attempted to serve G-d in a way that was not commanded. Possibly this was the effect of the seven day consecration process in the previous chapter (chapter 8) that it prompted them with a desire to serve G-d more than what was commanded. This is a very common desire with the religious personality that one wants to do "more," and the seven day consecration process was an intense religious experience that could very likely have inspired this feeling.
After discussing this idea in my synagogue, a friend, Paul Kloot made the interesting point that Aharon's action might have influenced his sons Nadav and Avihu since undoubtedly they were in close vicinity to Moshe and Aharon. Paul wondered that if they had not realized that Moshe had to rectify their father's actions, then Aharon’s action might have led them to think that it was ok for a person to serve G-d based on one's own ideas.