The Batcave is actually a nearly depleted Lazarus Pit and Bruce doesn't know it.
The Wayne family used to be aligned with Ras al'ghul and made their fortune being Ras' legit front. Bruce's grandfather severed ties with Ras amicably before Thomas Wayne was born, over some immoral merchandise Ras wanted the Wayne's to acquire(prisoners to test a deadly strain of flu we know as the Spanish Flu). Ras didn't mind the split because the Wayne's were basically redundant at this point. With the rise of corporations Ras had something equally as immortal as him self that could always serve his needs. That and the Wayne Manor Lazarus pit had become of no more use to him after an earthquake destroyed most of the substance with in.
Bruce's grandfather never told his son Thomas about the Lazarus pit or Ras. Thomas still managed to discover the remains of the pit and the life giving properties. He used the weakened Lazarus pit vapors in his surgeries to great effect. Becoming a highly regarded and almost miraculous surgeon. Who could pull of surgeries that were expected to fail with the patient's death. His fame helped him bring Wayne Enterprise to the most prominent and successful corporation Gotham. The Wayne's notoriety had not escaped the notice of Ras who had deduced the secret of Thomas' miracle surgical performance. So Ras set forth a plan in motion to make the great Thomas Wayne's death look like a random act of street crime. Ras had also planned to portray himself as forgotten older brother of Thomas, who had been given up for adoption at birth. Ras would come in claim Wayne Enterprises and the orphaned son Bruce to be his heir apparent. His plan was foiled by a British Intelligence officer named Pennyworth.
Alfred had been on Ras' trail for years during the course of investigating a terrorism plot from Northern Ireland. Alfred had taken up a cover identity as the Wayne's butler after learning of the connection between the Wayne family and Ras. Alfred fought off Ras' attempt at taking over the Wayne name in a finale battle in the cave under the Manor. He was badly injured and took a risk and immersed him self directly into remaining Lazarus pit substance. While it wasn't potent as a normal pit is. It healed his injuries and froze his aging, but left him appearing much older then he was at the time of the battle. His final assignment after recovering was given to him. He was to remain under cover as the Wayne butler and guardian of young Bruce so he could maintain watch over the pit and the workings of Ras.
Alfred had become very attached to Master Wayne while raising him. When Bruce because Batman, Alfred afraid for the life of someone he considered a son, guided Bruce to set up the Batcave in a certain spot. The part of the cavern system under the Manor not only had natural secluded exits but also filled with Lazarus pit vapors which would help Bruce heal faster. Bruce is so far unaware of the vapor and its positive effects on his health.