Being a fish, the only way to battle the boss is to bring Link underwater. What he must do is attack the glowing ball that is placed above his eyes with your swords, bows, or whatnot. If you're quick, you'll be able to quickly kill the Angler Fish just by attacking that glowing ball.
The boss also appeared in the video game The Legend of Zelda: Oracle of Ages, where it appears in Jabu Jabu's belly as a mini-boss. Rather than fighting it in the water, however, the fight will take place on land, where it'll jump around attacking you.