This passage demonstrates an approach common to authoritarian regimes: the establishment of one universal ideological perspective to guide all of society's functions and infrastructure.
By isolating the scripture from those the administration of Gilead wants to oppress, they force oppressed classes to trust in the authority of those who do have the ability to deliver the scripture to them. This perceived authority reinforces the power of the ruling class, and it also elevates those who possess it to a perceived spiritual tier above those who can't access the material firsthand. In a sense, they might be seen by those underneath them as "closer to God." This would make the very pious members of Gilead much more likely to obey and respect them without question.
This authority and uniformity also cuts down on conflict and debate over the source material, which—if you're the oppressor—is especially important of a book as broad and multifaceted as the Bible. Throughout history, religious scholars, theologians, and secular readers alike have interpreted the text an infinite number of ways, and there still isn't a formal consensus among them. One might glean very different information from a feminist analysis of the Bible than someone else would from a Marxist interpretation, and both would probably be different still from the conclusions drawn by a utilitarian.
In a society like Gilead, where the uniform and rigid interpretation of information is key to the oppression of the masses, open dialogue is a liability—if the disenfranchised have the freedom to thoughtfully examine the material guiding society's rules, they may challenge the state's interpretation of that material. If they do, they may object to the legal and logistical implementations of that interpretation. If they do that, it becomes much harder for the state to maintain order—the people might resist in small ways, or they might organize a formal resistance.