Basically what the author is trying to say is that you don't want to c-bet all your hands, because you will have to fold to a shove a lot of the time. Your opponent can abuse you this way. He proposes to use hero's overpairs as bets, since they will not improve on many turns. The hand strength will rarely change, so there is not much value for hero in seeing a turn. To balance out this value range we want to bet the hands that have the least amount of showdown, like JTs, 98s. Ace high hands like AK and AQ have showdown value and can hit a pair or a flush/flush draw more often than an overpair. They benefit more from seeing a turn and maybe getting to showdown, so they should check.
So to get to your question. The author actually proposes to check a lot of ace high hands, because like you said, you have showdown value and want to get to the river. However your overpair hands are very strong. They will rarely be beat, so hero wants to bet for value.
I agree with the author's strategy here, however I would advice to sometimes check aces or kings to have a stronger checking range.
EDIT: so in short (and we can apply this to a lot of situations): bet your strongest and weakest hands to get value and folds respectively. Check your mediocre hands to get to showdown and avoid playing a massive pot, but include from time to time a very strong hand to strengthen your check back range.