To obtain the Angler Key, Link must defeat a Desert Lanmola in the Yarna Desert, located in the south-east of Koholint Island. Accessing the Desert requires Link to bring Marin to the desert entrance, to awaken the walrus blocking the path by singing Ballad of the Wind Fish.
If Link is not quick, the key may fall into the cave below after defeating the Lamola, but it can still be grabbed after following it down. Once he has the Angler Key, he can use it on the keyhole at Level 4: Angler's Tunnel, causing the waterfall to try up, revealing the dungeon entrance. He cannot walk straight from the keyhole after using the key to the entrance, he must jump down from the original waterfall source in Tal Tal Heights.
Angler Key sprite from Link's Awakening DX