Besides keeping your vehicle more level, a sway bar also limits the lean of the body. This is needed because it means that when you take a quick set into a turn, the body isn't still moving sideways after the tires at their limits. Otherwise you turn in quickly, the tires grip and then the body finally finishes leaning, when it stops, the tires loose their grip.
A sway bar limits camber changes. The camber is the angle that the tire leans in or out at the top relative to the chassis of the car. The camber directly impacts the angle at which the tire cross-section meets the road and thus controls lateral grip. As the suspension compresses the camber angle generally changes relative to the chassis. With a normal Macpherson strut that hasn't been lowered, the camber goes from positive to more negative as the lower A arm swings out straight, and then back to positive as it swings up. That swing up into positive camber is bad. At that point the chassis is already leaned over so the tire may be starting to roll onto its sidewall. Changing the camber even more positive just nasty. A big sway bar will prevent the body roll in the first place, and prevent the suspension compression on the outside that causes the positive camber change relative to the chassis.
A sway bar will transfer lateral grip from one end of the car to the other. Racers exploit this every time they go on the track. Their spring rates are often so high, the cars so low, and their suspension travel so little, that the whole camber and body lean problem is already a non-issue. The car doesn't lean much with 500 lb springs. They use their bars to change the balance of the car. A big bar on the front, increases rear lateral and motive traction. A big bar on the rear increases front lateral and motive traction. If the car is under steering, decrease front bar size, or increase rear bar size. This increases front lateral grip and decreases rear lateral grip giving the car a more neutral to over steer feel. Reverse the process for too much over steer.