In his speech, Elie Wiesel is trying to persuade people that they need to care more about other people.  He is saying that they have to stop being indifferent about the terrible things that happen to other people in the world.

If that is what he is trying to do, it makes sense to refer to the "vanishing" century.  He is trying to say that genocide and indifference to genocide are things from the last century.  It is saying that indifference is something that should vanish along with the last century.