Just because a wheel is touching the ground, doesn't mean you have traction, but sway bars help distribute the weight of the vehicle more evenly.
Think of this scenario:
Your drivers side wheel is down in a rut while the passengers side is up on the bank. Whether you have a sway bar or not, you can usually have both tires touching dirt. But disconnected, the front end weight isn't as evenly distributed between the too front tires. Most of the weight could be over the wheel on the bank, and the drooped wheel is basically just resting it's weight on the ground. With a light rate sway bar like the Antirock, the bar will pull the body of the vehicle a bit more parallel with the frame. With the body tilted a little more into the rut, more weight is over that tire (not exactly 50%, but more than just the weight of the wheel in the discoed case).
The more weight over a tire, the more traction that tire has as it is forced into the ground harder. So the AR gives you two tires that have decent traction versus one tire with good traction. That gives you a better chance of finding traction.
It also helps with soft terrain by making it harder for one tire to dig down and high-center the rig.