We would love Nerd Words to be a game that families, classes, and gaming groups play over and over. The purpose of these boxes is to allow groups that play this game a lot to have a semi "legacy" version of the game so they are not reusing the same subset of Science Terms over and over or always choosing the easiest terms.
When a Science Term is used, players mark one of the boxes next to that term with a pen or other permanent writing tool. Usually, in a guessing game like this, you would try to separate cards you've used from ones you haven't seen yet, but in this version, all the cards are shuffled back into one deck again. If a card comes up in the next game that has a mark on it, the Clue Giver must use one of the Science Terms with the fewest marks next it, forcing all terms to be used an equal number of times, and easier terms (or terms already used) do not come up again until all other terms have been used.
You don't have to play this way, but it does ensure that you end up using--and learning--more terms in your playthroughs!