Here's why:
* Adverb placement: The adverb "silently" should modify the verb "remain". In this case, it should come directly after the verb: "She wanted him to remain silent."
* Adjective vs. Adverb: "Silent" is an adjective, while "silently" is an adverb. Since we're describing the manner of remaining, we need the adverb form.
Corrected sentence:
She wanted him to remain silent.