What are the connotations off "Vayitav be'Einav"?
Rashi: Moshe admitted that Aharon was right. 1
Seforno: He rejoiced over his brother Aharon and his son's straight line of thinking, that they got it right and taught him the truth.
Targum Yonasan: He was pleased with Aharon's reply, and announced in the camp that he had forgotten the Halachah and that Aharon his brother had reminded him.
In spite of the shame, he admitted that he had been taught what Aharon said and that he forgot it (Rashi).
What caused Moshe to err?
Rashi (from Vayikra Rabah 13:1): Because he was angry with Elazar and Isamar, he forgot. 1
And anger causes a person to make mistakes (Rashi, Bamidbar 31:21). Hadar Zekenim (Shemos 16:20) - also in two other places, Moshe became angry and forgot. He got angry at people who left over Manna, and forgot to tell Yisrael that they will get extra on Friday for Shabbos (ibid). When he got angry that the soldiers did not kill Benos Midyan, he forgot to tell the people to Kasher the Kelim taken from Midyan (Bamidbar 31:21).
QUESTIONS ON RASHI
Rashi writes that because Moshe was angry, he forgot. Before he forgot, why did he get angry? Aharon and his sons did properly!
Rashash (on Vayikra Rabah 13:1): This is like the opinion (Zevachim 101a) that it was burned due to Tum'ah; if not, they could have eaten it at night, after Torah Aninus ceased. Moshe, unaware of the Tum'ah, was angry why they burned it, and did not leave it to eat at night. He thought that amidst grief, they were negligent and it became Tamei. Amidst anger, he forgot that they may not eat it during the day, and asked why they did not eat it.
Toras Moshe (19): They should have eaten a k'Zayis immediately after Haktaras Eimurim. 1
Toras Moshe: Aharon answered "ha'Yitev b'Einei Hashem" - the Zakai atones for the Chayav. Until the decree against me (that my sons die) was [partially] fulfilled, I was not Zakai; it would not have atoned properly. Hashem caused that we not eat promptly, for Yisrael did not deserve the full Kaparah! Moshe accepted the answer.
Rashi writes that Moshe was not ashamed to say 'I did not hear.' What is the Chidush? This is a Midah of Chachamim (Avos 5:7)!
Riva, Rosh (19), both from Zevachim 101a: Moshe did not say, due to shame, ' I did not hear.' Rather, he said 'I heard, and I forgot.' 1 This is the intent of Rashi, even if it is not explicit in Rashi's words.
Toras Moshe: Moshe did not hear explicitly, but he was shown what every Chacham will be Mechadesh, He himself was Mechadesh the law, but Aharon went deeper to distinguish Kodshei Doros from Kodshei Sha'ah. Moshe admitted that he did not fully understand his own Chidush; he was not concerned for lest people say that one cannot rely on him. "It was good in his eyes" to share with Yisrael Pilpul (deep investigation) which was given to him alone.
Shir Ma'on: The Chidush is that he recognized that Aharon was right - "Toras Emes Haysah b'Fihu"(Mal'achi 2:6) - and admitted, and he was happy about this.
Moshav Zekenim: In Shabbos (119b), we find that it is a greater shame to say 'I never learned' than 'I learned and forgot'! There, one should have learned from a Rebbi. Here, if Hashem did not tell Moshe, he is not to blame for not knowing! Divrei David - there, to say 'I never learned any Torah' is a great shame. Here, to say 'I never learned this law' is not a big shame. Alternatively, here, to say 'I did not hear explicitly. I reasoned this way, and Aharon convinced me to retract' is not shameful. This does not apply there.