If we assume that this idea is true, it is true because it will be civilization, rather than things like ideology or geography, that will cause conflicts in the future.
In the past, conflict was often caused by things like ideology (communism v. democracy) or by geography (German-English or German-French rivalry). Today, however, those sorts of factors do not seem to be the causes of conflict. Countries like Germany and France are no longer enemies and there is no longer a Cold War to divide the world ideologically.
So now, the conflicts will come (if Huntington is right) from differences between "civilizations." The US, for example, will come into conflict with China and with Iran because these countries all have different worldviews and different cultures.
This can help us understand where conflict will arise in a post-Cold War world.