You will seek to have issues removed from the democratic sphere and transferred to the domain of judges and courts, where one can achieve one's goals without having to win majoritarian support. Attacking civility produces a polarizing effect in both the political and legal worlds. In the former, the absence of civility will help throw a wrench into the political process. Encouraging rudeness, nastiness, and personal attacks will make actors in the political system less likely to compromise and less willing to engage in meaningful deliberation. The different parties will be less likely to reach a consensus that blends different, opposing views.
Incivility also discourages individuals from taking an active interest in public affairs.
Furthermore, if the political system becomes stymied, there will be more pressure to take the issue to the courts. In the legal world, acting uncivilly will discourage parties from settling cases, or even agreeing on uncontroversial issues of law and fact. To think of it in a different way, civility is necessary if one believes in the law as a neutral, autonomous system in which there are right and wrong answers. Civility then is but the natural functioning of a legal profession in which we are all servants of that higher, nobler master, the Constitution and the law. The lawyer on the other side, or the judge, is not the enemy, but a fellow traveler on the journey toward discovering the correct legal answer.