The Recorder has three known uses. Also, only two of these three is required to complete the game; the last one is completely optional.
DIGDOGGER HATES CERTAIN KIND OF SOUND
The first use of the Recorder is also the main one. If played while Link is fighting a Digdogger, the Digdogger will shrink into a much smaller version of itself. Link can defeat this mini-version of Digdogger with his Sword. It is impossible to defeat Digdogger without using the Recorder to weaken it first.
Level 7 Entrance
The second use of the Recorder is very specific and is only needed once. On the western side of Hyrule, there is a Fairy Fountain that does not house a Fairy. If Link plays his Recorder while on this screen, it will reveal a secret; the entrance to the seventh dungeon of the game, Level 7: The Demon! If Link ever needs to re-enter this dungeon at any point in the game, the Recorder will work everytime.
The final thing the Recorder can do for Link is this: when played in the Overworld, Link will be warped to the entrance of a dungeon that he has previously defeated. The destination is determined by which direction Link is facing while using it, making the Recorder a very effective way of travelling across the overworld.