The cave was discovered long before Bruce Wayne started using it as his base of operations. During the Civil War, Bruce's ancestors used the caves to get escaped slaves to freedom. Eventually, Bruce rediscovered the caves and began to use it as is home base. Bruce and Alfred refered to it as "the Batcave."
The Batcave could be accessed through a hidden stair case behind a Grand Father Clock or by an elevator located behind a book shelf in the mansions library. To open the Grand Father Clock Bruce Wayne must turn the hands to 10:47 which is the time that his parents were murdered.