Why does the stereochemistry of the hydrogen to be removed must be anti to that of the leaving group for E2 reactions?
This is because E2 elimination occurs most often in the anti (periplanar) geometry. This arrangement would allow the molecule to react in the lower energy staggered conformation (the base and the leaving group is farther from each other). If you have a syn (periplanar) arrangement it would result to a less stable conformation. We don't want that.