As others have pointed out, the title of this film (and novel) is related to an anecdote that Clarice Starling shares with Hannibal Lecter.
When she is young and sent to stay with relatives, Clarice hears the screaming of lambs that are going to be slaughtered. She tries once to save a lamb and run away with it but fails.
As an adult, she admits to a recurring nightmare that features the wailing cries of these lambs.
There are at least two ways to read the significance of the title in light of this anecdote. First, the lamb is a common symbol of innocence and of sacrifice. However, in order for a lamb to be a "sacrificial lamb" it has to killed under ritual circumstances or for ritualistic purposes. This is the case with the Senator's daughter, who is held captive by the serial killer Buffalo Bill. He is engaged in the ritualistic killing of young women in an effort to transform himself, using the skin of their bodies to create a new skin for himself.
In this way, the notion of the sacrificial lamb is applicable to the narrative in The Silence of the Lambs. Reading the conflicts of the plot against this symbolic backdrop, we can argue that Clarice is engaged in an effort to stop the ritual killing of the sacrificial lamb (the Senator's daughter) and in doing so undo the cult of celebrity and of fear that Buffalo Bill has attained in the news.
A more straight-forward reading is to see Clarice's quest as a personal one wherein she finally succeeds in saving the innocent figure of "the lamb" and so quiets the screams in her nightmares. She is on a quest to redeem her own sense of power -- the power to be effective, to be strong, to save others.
The layers of meaning in the title can also be seen to reflect ideas of childhood trauma stemming from sexual abuse that occur in Clarice's past and that also apply to other characters and anecdotes in the narrative. Clarice is tasked not only with saving an innocent woman but with developing into a potent woman who will no longer exist at the mercy of male authority. This motif connects to the Miggs character, to Buffalo Bill, and to Clarice's personal history.
"Although she is not unaffected by the events of the novel, at the conclusion of the novel she has earned the "silence of the lambs," which at least for the time being represents peace" (eNotes).
No longer a lamb, in a manner of speaking, Clarice grows into an adult. She is not innocent, but rather is redeemed. She is not haunted anymore by a sense of powerlessness as she has fulfilled her quest and proven her capabilities.