Havolim says all that needs to be said in terms of the lesson for us to take away from, "Vayidom Aharom," Aharon's silence in response to the death of his children.
A bit of Navardok mussar: Aharon is praised for his silence, but what could he have said? Surely Aharon was no less righteous than the many tzadikim through the ages who faced trials and loss without even thinking of questioning Hashem. Why is Aharon's silence special?
The answer is that there is something Aharon could have said: "Kol d'avid Rachmana l'tav avid" -- the qunitisenntial declaration that all that Hashem does is for the good, whether we understand it or not. Chazal tell us that just as one must say a blessing on the good things in life, one must also say a blessing on the not so good. This is our expression of trust that ultimately, whatever Hashem brings upon us, is for the best.
But why then was Aharon silent?
When we say our "Gam zu l'tovah," it is a response to the perception that something bad has happened. We don't want to question G-d, we don't want to blame G-d, so we say to ourselves that in reality it's all for the best. Our words are an attempt to bridge that gap between our perception of pain and the reality of G-d's goodness. Yet, as sincerly as we may utter those words, there is yet a higher madreiga. That higher madreiga is to not even perceive that which others call "bad" as being bad at all -- to be so filled with the knowledge that G-d alone controls the world and everything He does is for good as to be completely incapable of seeing pain, suffering, the absence of good. When there is no gap between the reality of Hashem's goodness and our perception, there is no need to utter a "gam zu l'tovah" as a reminder of G-d's benevolance.
This is the silence of Aharon haKohen.