“Balance” here is referring to overcoming imbalances through unilateral movements, thus “balancing” out the body. Work on holding yourself to the same standards for each side – i.e., if you can squat all the way down and touch the floor while balancing on your strong leg, don’t let yourself simply bend over, rounding the back, in order to touch the floor when balancing on your weak leg.
“Close Grip” = hands should be directly above your shoulders after un-racking the bar (DON’T grip the bar narrower than your shoulders, as this puts undue stress on the wrists). When performing the close grip bench, keep the elbows tucked in towards the body, and concentrate on keeping the eccentric portion of the lift nice and controlled, while making the concentric portion explosive, driving the bar up away from your chest as quickly as possible.
There were two requirements for this workout; First, you must jump continuously through-out the 30 seconds of work, which led to the other requirement; Second, use a box lower than you would usually jump on to help facilitate the continuous jumping requirement.