First you need to confirm that the socket will actually work with your headphones. Make sure that the ground connection of the headphones (either the sleeve or the ring right next to the sleeve, depending on the type of headphones) is serviced by a connection.
Second you need to wire one or both of the tip and the first ring to the Arduino's output that you are using to generate audio. This should be done through a capacitor to block the DC voltage and only pass the AC waveform.
simulate this circuit – Schematic created using CircuitLab
Ideally, though, you should first pass the signal through a small headphone amplifier otherwise you risk drawing too much current out of the Arduino's GPIO pin.