In the "correct" answer, "silentaj" describes the boys (in other words, is an adjective). Using "silente", which was my first choice, "silente" is an adverb and describes/modifies the verb - tells how they rested.
Kial ne "silente" ? Tiel estas pli logika!
Well, one describes the way that they remained, and one describes the boys after the remaining.
"the way they remained" is clear. But how "after the remaining"?
I'm sorry, but I can't understand your point of view.
The second one seems odd to me. That's why I would suggest "La knaboj restis silentaj."