What was Moshe referring to when he said to Aharon "Hu asher Diber Hashem 'bi'Kerovai Ekadesh' "?
Rashi (from Zevachim, 115b): He was referring to Hashem's statement to Moshe (in Shemos 20:43) "v'No'adti Shamah li'Venei Yisrael, v'Nikdash bi'Chevodi" - as if it had written 'v'Nikdash bi'Mechubadai' (and it [the Mishkan] will be honored by (the death of) My honored ones.' 1
Ramban, Moshav Zekenim #1: It is not referring to any prior statement of Hashem, but is rather a direct reference to the death of Nadav and Avihu, as if to say 'This is what Hashem decreed, saying (in His heart) 2 that He will be sanctified via those who are close to Him - that they should not breach His Kedushah, and that He will be honored via all the people - that they should behave respectfully in His Mishkan.
Rashbam: When Aharon's sons died and he wanted to leave the Avodah and mourn for them, Moshe informed him not to mourn, not to cry and not to desecrate the Avodah by leaving it, because Hashem had told him that He wished to be sanctified by the Kohanim Gedolim 3 not leaving the Avodah - by not mourning, but continuing to serve as if nothing had happened. 4
Moshav Zekenim #2: Hashem had said "v'Gam ha'Kohanim ha'Nigashim El Hashem [Yiskadashu Pen Yifrotz Bahem Hashem" (Shemos 19:22. Even though that discussed ascending Har Sinai), even a Chacham's conditional curse is fulfilled (even if the condition was not fulfilled), all the more so Hashem's curse!
Moshe was now informing Aharon that he knew that something like this would happen, only he thought that the victim would be either himself or Aharon. Now he knew however, that Nadav and Avihu were greater than they were (Rashi). See Rashbam's objection to this explanation. Targum Yonasan however, explains that Hashem was only issuing a warning that if the Kohanim are not careful when performing the Avodah, they will be burned by means of a fire that comes from Hashem.
Regarding this, Aharon's sons were considered like Kohanim Gedolim [during the Milu'im].
As the Pasuk specifically states in 21:10-12; as that Parshah preceded this one ('Ein Mukdam u'Me'uchar ba'Torah [Rashbam]).
Why is Hashem's esteem enhanced when He carries out Din on Tzadikim?
Rashi: Because people will then say that if Hashem performs judgment on Tzadikim, how much more so will He do so on Resha'im. 1 And so the Pasuk writes in Tehilim, 68:36 "Nora Elokim mi'Mikdashecha" - as if it had written 'mi'Mekudashecha' (Hashem is awesome when He performs Din with His holy ones).
Ramban: And they will know that He dwells among them. Da'as Zekenim - they will guard themselves from sin and fear Him.
What does "al-P'nei Kol ha'Am Ekaved" refer to?
Ramban: Refer to 10:3:1:2.
Rashbam: When they see that his sons died yet he refrains from mourning and continues with his Creator's Avodah (Refer to 10:3:1:3).
What does "Vayidom Aharon" mean?
Ramban #1, Moshav Zekenim: He stopped sobbing loudly.
Ramban #2: His eyes stopped shedding tears. 3
Rashbam: He refrained from crying and from mourning. 4
Seforno: It means that Aharon was comforted by the Kidush Hashem that resulted from his sons' death.
Moshav Zekenim: It similarly says "va'Yidom ha'Shemesh" (Yehoshua 10:13) - when a Tzadik dies, even the sun ceases to sing Shirah.
But accepted the decree with love (Rashi, 11:2).