Those stairs are actually the best secret in the whole game!
In order to reach those stairs you have to swim in the moat surrounding Hyrule castle until you have found the secret entrance at the back of the castle. The entrance is almost perfectly in the middle of the back of the castle except it is one tile to the left. You will know when you're close to the entrance since the back wall is indented at this point (for the length of the tower in the middle of the castle).
The moat is deep water. You need a certain item in order to swim in deep water. You can obtain this item-
at Lake Hylia by waking up the Zora Prince who is asleep in a cave. You can wake him up by bombing the Zora ghost that talks. Defeat the ghost and the Prince will wake up! After you wake him up he will reward you with the Zora Flippers.
Chances are that you already have this item if you are at this point in the game!