This is a great question. Not much is known about the Sibylline books in Roman history. The books were destroyed in 83 B.C. when the temple of Jupiter Optimus Maximus was destroyed by fire. However, in 76 B.E. Gaius Curio, a consul, sent a commission to Erythrae to obtain Sibylline oracles. This commission returned with about 1,000 verses. They also probably got some more fragments from other places.
According to tradition, an old woman came to Tarquinius Superbus, the last king of Rome, and offered him nine rolls for an exorbitant price. When Tarquinius refused to pay, she burned three of them and asked for the same price. Tarquinius ridiculed her. She burned three more and asked for the same price again. At this point, Tarquinius thought twice and purchased the scrolls.
The Sibylline oracles were under the guidance of one of the priesthoods of Rome, the quindecemviri. They would look into the books and discern the will of the gods. Usually the senate would ask the quindecemviri to look into the books after some unusual event that scared them. In turn, the priests would offer their opinions on what expiations should be done to restore peace with the gods.