It's the result of a weak or decentralized government. One faction within the government decides they'll start burning books. People resist by force, and another faction within the government grants them the power to do so legally, knowing they would regardless.
Not very realistic, but similar things have happened before. One, for example, is the federal government using the military to enforce desegregation in the US, while some southern states attempted to use the national guard to prevent it. It was never carried to the extremes in the anime, but it easily could have.