It is Ralph who realizes that "The world, that understandable and lawful world, was slipping away." Ralph's realization can be found in chapter five and on page eighty-two depending on the edition you're using. This quote is significant because Ralph is coming to realize that the order, customs, and civilized life is slowly being replaced by something wilder, feral, and more chaotic.
In Chapter Five of Lord of the Flies, Ralph is struggling as a leader and finding that the sense of structure within the group is slowly dissolving. He calls a meeting with the conch in order to lecture the boys about their ignorance of the rules and their neglect of their expected duties (building shelters, fetching water, using the appropriate toilet area, tending the signal fire, etc.). Ralph believes that monsters do not exist on the island, and Jack and Piggy back up his claim.
When Simon suggests that there really is a beast, Jack rapidly changes "sides" and decides that he will form a group of hunters to track it and kill it. Jack flees with a group of the boys, who have now begun to dance and chant wildly.
Thus, the narrative description intervenes during this shift in power to suggest that, "The world, that understandable and lawful world, was slipping away." The boys are losing contact with both reality and their sense of civilization as human beings.
This statement is not part of the dialogue, so nobody says it aloud. However, it is part of the narration of Ralph's thoughts. Since the boys landed on the island and Ralph was elected chief, he has been pursuing two objectives: having a functioning temporary society and getting rescued. He developed some practical solutions that would have been quite successful if only the other boys had cooperated with the rules. He has made rules about shelter, safety, and sanitation. But most importantly, he made a rule about keeping a signal fire burning at all times, and he delegated to Jack the task of making sure the signal fire was constantly manned.
Unfortunately, Jack neglected that responsibility and drew the fire-keepers away into a pig hunt at the very time a ship passed the island. If that had not been the case, the boys would probably already have been rescued by the time the meeting gets to the point at which Ralph thinks this thought. In his outrage and disappointment, he calls a meeting at dusk to clarify the rules, and he makes the mistake of trying to deal with the boys' irrational fears as it is getting dark. He then asks the wrong question of the group: "Who thinks there may be ghosts?" Many boys raise their hands in the affirmative.
At this point, he says, "I see," and thinks "the world, that understandable and lawful world, was slipping away." What had been so clear in his mind about how to live on the island and how to get rescued seems to be nothing more than a phantom in the face of all the boys who don't have the same priorities as he does. When one finds himself alone or in the minority opinion on something, it can be hard to maintain one's own convictions. That is the situation Ralph finds himself in when he has this thought.