Research suggests that the swinging is basically passive. That is, the movement of the body when walking causes the arms to swing. Preventing swinging actually causes the body to use more energy, since you must counteract the natural swing. Several experiments suggest NOT swinging the arms causes the energetics of walking to go up by about 10%. 

Surely this is about your rotational stability around the vertical axis.

Consider you have just finished a step forward with the left leg. When you recover the right leg by accelerating it forward with the front thigh muscles, it pulls the right side of the body backwards (Newtons first law). Either you involuntarily turn slightly to the right, or you counteract the movement by a rotational strain through the left foot. (If you are running, the recovery of the right leg begins before the left foot touches down, so you can't do this.)

The correct dynamic response to this is to apply a reverse torque around the vertical axis using the arms: when pulling the right leg forward, you push the right arm back and the left arm forward. Apart from stabilising your vertical rotation, this enables faster leg recovery and therefore faster walking and running.

It's noticable for very tired runners that they lose the arm movement first. Instead, they tend to push their heads forward to increase their moment of inertia around the vertical axis. Also, they appear to turn their face side to side to compensate for body rotation.