How did the church save civilization, as some historians have suggested, in the western world?
One could debate exactly what is included in "civilization" or the "western world." Because your tags include reference to "history of the catholic church in western civilization," the response becomes easier to define.
The Roman Catholic church became the preserver of much of the structure upon which western civilization was built during the so-called Middle Ages - the period of time ranging roughly from the fall of Rome and the Roman Empire in 476 CE to the Renaissance in the 1400s.
The Church hierarchy of priests, bishops and other leaders served as advisors to the feudal lords and kings who exercised whatever political control there was in a given area. Any education that was delivered during that time came through the Church, as children were trained to assume roles of responsibility within religious organizational systems. Some monastic orders preserved the history and traditions of previous civilizations as a part of their mission to create the manuscripts that recorded the historic writings saved from destruction.
The main argument for how the Church saved civilization is based on education. This argument holds that the religious in the Middle Ages were the only large group of literate people.
During the Middle Ages, there were many religious houses (monasteries and the like) all over Europe. Many of these religious houses did a great deal of work with books. They copied books that other houses had and added them to their own libraries. By doing this, they preserved many classic works that would otherwise have been lost. They also ensured that there would be a relatively large number of people who were literate and who could form an intellectual class that would help to ensure that Europe remained a true civilization.