In spite of being warned against sharing personal information with Lecter, the FBI trainee Clarice Starling agrees to a quid pro quo with Lecter who offers information about the serial killer “Buffalo Bill” but requires personal information in exchange. In one of their conversations the FBI trainee tells Lecter of her younger years when she was adopted and raised on a farm. One night she heard the sheep screaming and on going to check on them she found them being slaughtered. She then opened the barn and tried to run away with one of the sheep but was caught and the lamb slaughtered. She admits that she often hears the screams of the lambs when she sleeps. Based on Lecter’s psychoanalysis he tells Clarice that she is dedicated to saving the Senator's daughter because she believes this will stop the problem she has at night. At the end of the movie Lecter calls Clarice and asks if the lambs have stopped screaming; this is after she rescues the Senator's daughter. This forms the basis of the movie's title.
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.
True, it is the title of the movie, but it is also a novel by the same name by Thomas Harris.
It is called "The Silence of the Lambs" because lambs, when they are led to slaughter, go quietly and without making a sound. Lambs are also perceived to be innocent and trusting, as are Lector's unfortunate victims in both the novel and the film.