I just saw The Silence of the Lambs (1991). So what exactly does "silence of the lambs" mean? It was not clear to me.
It refers to a conversation between Clarice and Hannibal Lecter. In order for Lecter to help with her investigation, he demands she tell him information from her personal life, to satisfy his curiosity.
Her story begins with her upbringing in West Virginia with her father. He was later shot. She lived with her mother, but was sent away to live with her uncle as she couldn't afford to keep her. Her uncle owned a sheep and horse farm, and she learned one night that the lambs on the farm were slaughtered - in horror, she ran away.
This story is covered in the script. Note the highlighted part towards the end:
Clarice (voiceover): Lambs. The lambs were screaming...
Dr. Lecter: They were slaughtering the spring lambs?
Clarice: Yes...! They were screaming.
Dr. Lecter: So you ran away...
Clarice: No. First I tried to free them... I opened the gate of their pen - but they wouldn't run. They just stood there, confused. They wouldn't run...
Dr. Lecter: But you could. You did.
Clarice: I took one lamb. And I ran away, as fast as I could...
Dr. Lecter (voiceover): Where were you going?
Clarice (voiceover): I don't know. I had no food or water. It was very odd. I thought - if I can even save just one... but he got so heavy. So heavy...
Clarice (voiceover): I didn't get more than a few miles before the sheriff's car found me. The rancher was so angry he sent me to live at the Lutheran Orphanage in Bozeman. I never saw the ranch again...
Dr. Lecter (voiceover): But what became of your lamb?
Dr. Lecter (voiceover): Clarice...?
Dr. Lecter: You still wake up sometimes, don't you? Wake up in the dark, with the lambs screaming?
Dr. Lecter: Do you think if you saved Catherine, you could make them stop...? Do you think, if Catherine lives, you won't wake up in the dark, ever again, to the screaming of the lambs? Do you...?
Clarice: Yes! I don't know...! I don't know.
The Catherine referred to is the U.S. Senator's daughter who is kidnapped and who Clarice is trying to locate.
The implication is that by saving Catherine, by saving this innocent girl (innocent like the lambs), she'll no longer be woken by the howling of the lambs in her dreams - the lambs she was too weak to save. Instead, she'll have saved the innocent and she'll be able to sleep in peace.
The title comes from the very last line of the book. As Andrew Martin states, early in the book, Starling and Lecter have a conversation about her having nightmares with screaming lambs and that she thinks the nightmares will stop if she can save Catherine
The book finishes with Clarice Starling having saved Catherine, and true to the conversation, her nightmares have stopped.
But the face on the pillow, rosy in the firelight, is certainly that of Clarice Starling, and she sleeps deeply, sweetly, in the silence of the lambs