After Aaron's sons, Nadav and Avihu are consumed by G-d's fire for having brought their own, spontanious offerings, Moses is quoted in the verse 4 as saying:
This is what was meant when the LORD spoke, saying, "I will be sanctified by them that come near me, and before all the people I will be glorified"
How are we to understand this verse? Where and when did G-d speak the words? These questions were debated amongst the Jewish commentaries. Most agreed that Moses was referring to a verse elsewhere in the Torah, but they disagreed about which verse that was.
Rashi (Rabbi Shlomo Yitzchaki, France, 1040-1105) quotes Talmudic and Midrashic sources which explain that Moses was referring to Exodus, Chapter 29, verse 43, where G-d explains the purpose of the Mishkan (tabernacle):
For there I will meet the Children of Israel, and it (the tabernacle) shall be sanctified by my honour
Rashi explains that the words and it shall be sanctified by my honour are a hint to Moses that in the future, on the day that G-d graces the tabernacle with His presence, the tabernacle will be sanctified by those who honour Me. Somehow, through the death of Aaron's sons, who were honouring G-d, the tabernacle was sanctified.
It is important to place Biblical commentary in it's historical context, and Rashi certainly was living in a time when sanctification-through-death may have been viewed differently to today. Not many commentators seem to be bothered by this sanctification-through-death concept, and most of them either quoted Rashi or provided the same verse in Exodus as an explanation of Moses' words.
Rashi's grandson, the Rashbam (Rabbi Shmuel son of Meir, France, 1085-1158) however disagrees. In his opinion, Moses's words can be explained by a totally different verse, one that hadn't yet been mentioned in the text. (Like Rashi, he did not believe the Torah was written in chronological order.) The verse is found 11 chapters later in Leviticus, in Chapter 21, verses 10-12:
And the high priest ... shall not go to any dead body, nor defile himself for his father, or for his mother; Neither shall he go out of the sanctuary, nor profane the sanctuary of his God; for the crown of the anointing oil of his God is upon him: I am the LORD.
In these verses, Moses is given instructions regarding the behavior of the High Priest, and is told that amongst other restrictions, should the High Priest have a death in his family, he must not abandon his Priestly duties in order to partake in a mourning ceremony. (This idea might also be connected to the Jewish custom of cutting short mourning periods for Festivals).
For the Rashbam, the Midrashic and Talmudic explanation of Moses' words to his brother were not valid, when compared to his textually oriented reasoning. Moses' words to his brother were a reminder that despite this awful moment in his life, his Priestly duties did not allow him to mourn at this time.
And this is how Aaron's response:
And Aaron was Silent
suggests that he did change course, and refrain from mourning in order to fulfill his priestly duties.