Psalms is all over modern culture, there's no question about that. But why? Maybe because of its ability to transcend the simplicity of the stories in the rest of the Bible—it digs deep into them, finding the commentary on faith that it's looking for. Yes, Psalms does occasionally do some rehashing, but more than anything else, Psalms is a book about faith, and how to deal with a skeptical world. That's a universal, and a powerful, message, and that's how Psalms has had such staying power.
Since illiteracy was much more common in biblical times, the Psalms would have been recited aloud at community luncheons, around the fire, and in the town square. People would memorize them as comforting lines in bad times, as admonishments for people they didn't like, and as a way of recalling their past together.
The writers lived in a world with political concerns just like ours. They were part of a larger world struggling to promote their ideas to an audience preoccupied with idol worship, gold, and survival. But of course, the writers were writing for God as much as for their own personal political projects.
Whether the Psalms were divinely inspired or written by the hand of man is irrelevant to the poetry, which has lasted for thousands of years. Can't argue with longevity, folks.