The Bird Key is located in a cave in the Tal Tal Mountain Range, behind a seemingly-impassable chasm. While it can be seen from a ledge, obtaining it requires Link to have revived the Flying Rooster from under the Mabe Village Weathercock with the Frog's Song of Soul. The Rooster can carry Link over the chasm to reach the key.
Once Link has the Bird Key, he can use it on the keyhole at Level 7: Eagle's Tower. This rotates the Tower, such that Link can walk up a flight of stairs and into the Tower.
Link's Awakening DX Bird Key Sprite