Contra-body movement simply means that the shoulders and hips are rotated in opposite directions. For a simple example, walk forward while letting your arms swing. You should feel most stable if your left foot and right arm go forward at the same time, and vice versa. If you try to walk forward with your left foot while moving your left arm forward as well, you will probably feel off balance and your walking will look closer to Frankenstein than Michael Jackson.
The following drill is designed to help get your upper body rotating in opposition to (or contra-) your lower body.
The Drill: Stand up straight with your feet side by side. Starting with your left foot free, do a triple step with a cross-and-side position. In other words, the first step of the triple will be your left crossing in front of your right, the second step will be recovering onto your right, and the third step will be on your left, alongside your right. Repeat with the other foot (right crosses in front of the left and then lands alongside the left).
As you do these triples, think about pulling back the shoulder when you take the cross step. In other words, when your left is crossing in front of the right, your left shoulder should pull back; when your right crosses in front of your left, your right shoulder should pull back. You can start by simply keeping your shoulders square to the wall in front of you; as that becomes comfortable, work on adding more twist through your shoulder and core so that the contra-body rotation becomes more pronounced.
(Obligatory footnote: I say shoulder to help focus on your upper body movement, but really the rotation should come through the whole side of the body. You should be able to do this drill while keeping your shoulder in frame. If you tend to roll your shoulder forward or pull it back without bringing the rest of your torso along, pause and reset your frame.)
Bonus Variation: When you are comfortable with contra-body rotation during cross steps, change the crosses to hooks. When your left hooks behind, you will need to rotate your left shoulder forward; when you hook your right behind, your right shoulder should go forward.