The Robe and the Song are both historical stories that the Council shares with the people in their society. When Kira gets the job to repair and restore the robe, she realizes that it's one way the Council has of controlling people—including her and the Singer, who produce the history the Council dictates.
By controlling the history of the society, the Council is able to control how people see their own society. It creates peace because people accept that things are going to be a certain way when they see that history has always gone this way, that events led to this moment, and that things are better now.
The Ruin Song explains that there was a fiery cataclysm that ruined entire cities and societies. This frames the current experiences of the people in Kira's society by showing them that they avoided that terrible fate. It also helps keep them from pushing for more technology and more autonomy; they don't want the same fate to befall them that happened to Toronto, Bogota, or Baltimore.
The history told through the robe also encourages the people to keep trying. As Kira is working on it, she recognizes a cycle to its history. Things build and decay and then get built again. As she works, she recognizes that all the work she's repairing and restoring is that of a past Thread-Weaver. This helps her realize that she can shape the course of history in the future along with the Singer and Thomas the Carver as long as they're willing to stand up to the Guardians who want to dictate the way history is shared and passed down.