Hi! You have found an error in the text! That line is a holdover from an earlier iteration of the rules.
The resolution procedure for the Prehistoric and Ancient Ages (step C: End of the Age) describes how the transition happens. The part that still reflects the misleading "Steward must pay AP for their creations to survive" line is if you roll a 7-9 you have the option of paying AP to allow your Avatar or People to survive to the next Age.
Does that make sense? It all works as intended if you ignore that line and just follows the step-by-step procedures for each Age. Sorry about the confusion!