As of Jun 17, 2014, the P3D/FSX/FS2004 client allows you to selected your audio output device from within the pilot client. Go to Start->Options-Voice, pick the desired device and click OK. The change takes place immediately.
EDIT: THE REMAINDER OF THIS POST IS OUT OF DATE, use the information above...
The PilotEdge FS2004/FSX client uses the default playback device at the moment, which is typically the speakers.
Here's how to get the audio to the headset for now:
1) Set the HEADSET to be the default playback device (and default
recording device)
2) Start PilotEdge (do NOT connect, since the sim isn't running yet)
3) Now set the SPEAKERS to be the default playback device again (this won't affect PilotEdge, it was already running and has already opened up the headset as its audio device).
4) Start FS2004/FSX, position the aircraft at the ramp
5) Connect PilotEdge to the network. You should now have ambient noises from the speakers and ATC audio in the headset.
Edit: ALTERNATIVELY, you can configure FSX to play its sound through the speakers, rather than using the 'default device'. If you do this, then you can skip step 3 each time.