They do two things. First, when your truck was lifted, the panhard ended up at an angle relative to the axle. Normally it's parallel to the axle. As the panhard travels through an arc it pulls the axle to one side. If you look down the side of your truck from the rear you can probably see that the rear axle is shifted slightly over to the passenger side while the front axle is shifted slightly to the driver's. The front moves the opposite way because the panhard on the front axle attaches to the axle on the opposite side of the rear. So, the bracket will put the rear panhard closer to parallel with the axle, centering the axle under the truck. This is purely aesthetic, it doesn't make the truck drive any differently to have the axles shifted to the side.
Second, as the suspension cycles, the panhard angle changes. Because it is already at an angle, the amount that the axle moves side to side as the suspension moves is magnified. So, when you hit a bump, the axle moves side to side a bit, causing the body to shift side to side. The lift bracket should minimize this movement. Same effect both on and offroad, although you will hit more bumps offroad. It takes a significant amount of suspension travel to move the axle an appreciable amount. My springs are probably stiffer than most so I don't notice the 'wagging' movement. Plus I just don't care as much as some people. My truck only goes wheeling, and comfort is third in line behind capability and my limited budget.
Unfortunately the front geometry is such that the panhard and drag link need to be at the same angle, so there's no easy way to fix this 'problem' on the front axle. You can get an adjustable panhard and center the axle for aesthetics, but that can lead to other issues if you run very large tires and stuff them into the wheelwells.
Second, as the suspension cycles, the panhard angle changes. Because it is already at an angle, the amount that the axle moves side to side as the suspension moves is magnified. So, when you hit a bump, the axle moves side to side a bit, causing the body to shift side to side. The lift bracket should minimize this movement. Same effect both on and offroad, although you will hit more bumps offroad. It takes a significant amount of suspension travel to move the axle an appreciable amount. My springs are probably stiffer than most so I don't notice the 'wagging' movement. Plus I just don't care as much as some people. My truck only goes wheeling, and comfort is third in line behind capability and my limited budget.
Unfortunately the front geometry is such that the panhard and drag link need to be at the same angle, so there's no easy way to fix this 'problem' on the front axle. You can get an adjustable panhard and center the axle for aesthetics, but that can lead to other issues if you run very large tires and stuff them into the wheelwells.

