Why was a perfectly ordinary bench so important to Karli?

The Bench is a story set in Apartheid-era South Africa. It begins with Karlie, a man who has been so indoctrinated into Apartheid racism that he has never considers the fact that maybe he is just as human as the whites who live alongside him. He attends a public protest of Apartheid, where whites and blacks are together on stage promoting the idea that all humans equal regardless of race. As he listens, he starts to believe them, and his entire view of himself and his world is changed.

These were new things and he, Karlie, had to be careful before he accepted them. But why shouldn't he accept them? He was not a colored man anymore, he was a human being.

Once he leaves, he goes to a train station and encounters a bench that is marked for Europeans Only. In Karlie's eyes, "it symbolize[s] all the misery of the plural South African society," just as the back of the bus symbolized African American misery for people like Rosa Parks. Karlie, as the narrator, perfectly encapsulates just why the bench is so important before he sits on it:

It was the obstacle between himself and humanity. If he sat on it, he was a man. If he was afraid he denied himself membership as a human being in a human society. He almost had visions of righting this pernicious system, if he only sat down on that bench. Here was his chance. He, Karlie, would challenge.

So, he sits on the bench and, a massive uproar begins. Eventually, a policeman comes and forces him off the bench, but Karlie is satisfied, as he asserted himself as a human being and won a personal struggle.
In summary, the bench is important because it symbolizes racism and the dehumanization that comes from institutionalized Apartheid. By sitting on it, he is able to overcome this, if only briefly.