The significance of the plaque on the bench near Bruno's house is to commemorate the building of the Auschwitz camp.  Bruno is at a phase of his experience where exploration becomes important to his identity. In exploring the plaque, there is a channeling of the lessons in which Bruno sees himself as a modern day Columbus or Vespucci:  "...men with such adventurous stories and interesting lives that it only confirmed in Bruno's mind that he wanted to be like them when he grew up." With this, Bruno focuses his energy into exploring the world around him.

Even though Bruno has been warned by his mother and father that going past such borders is expressly forbidden, Bruno's spirit of exploration guides him.  Prior to his exploration, he encounters the bench and studies the plaque that is on it:  "'Presented on the occasion of the opening of...' He hesitated. ''Out-With Camp,' he continued, stumbling over the name as usual. 'June nineteen forty."  The touching of the plaque fills him with such coldness, a reflection of the literal and symbolic experience of recognizing something that is commemorated with so much in way of death and human misery.  The plaque itself is significant because it indicates that people such as Bruno's father view "Out- With" as a marvel of efficiency. Bruno recognizes this plaque as marking a border of sorts.  This border has been one that has been denied to him, something that he is not permitted to explore.  In touching the plaque, Bruno makes clear that he takes his first steps in disobeying his parents, forging his own destiny and sense of exploration that goes with it.  In this, the plaque on the bench by Bruno's house acquires significance and meaning in terms of what it means for Bruno and the world around him.

Because The Boy in the Striped Pajamas by John Boyne is narrated by nine-year-old Bruno, there are some things in the novel which we (the readers) have to figure out for ourselves. One of them is the exact setting for the story, both in time and place, because Bruno does not always understand the correct names of things. 

Eventually we figure out that Bruno's father works for Adolph Hitler (Bruno refers to him as "the Fury") and gets promoted to Commandant after Hitler and Eva Braun come to Bruno's house for dinner. We also know that shortly after that Bruno's family has to move from Berlin to someplace Bruno calls "Out-With," where most of the story takes place.

Bruno is a boy who loves to explore, and he used to do it all the time in their house and neighborhood in Berlin. Here, however, he is not allowed to explore anything much outside of the house. He has been told time and again never to go near the fence or the camp which he can see through his bedroom window. It is such a tantalizing prospect for the boy that one day he has to find a way to explore. 

He starts right under his own bedroom window, and then he looks at the bench that he has seen from above. It has a plaque and he has never been able to see what it says, but today he intends to look. After making sure no one is watching him, Bruno goes to the bench and reads the small bronze plaque mounted on it:

"Presented on the occasion of the opening of..." He hesitated. "out-With Camp," he continued, stumbling over the name as usual. "June nineteen forty."

It is the last thing he looks at before he begins his forbidden exploration of the fence (in chapter ten) and his eventual discovery of his new friend, Schmuel.

The significance of the plaque is that this confirms that the camp Bruno has been watching from his bedroom (as well as all the activity of the soldiers in and out of their house) is, indeed, the infamous Auschwitz concentration camp. Even more, this is a place the Germans are quite proud of, as they commemorated its opening by mounting a plaque. The fact that it is on a bench is a bit disturbing, as well, for the presence of a commemorative bench suggests that the camp, at some point at least, was part of a kind of satisfying landscape for whoever was sitting in the bench, much like a sunset or a mountain view. Such a plaque at the entrance to the camp is one thing; a plaque on a bench rather implies a kind of satisfied accomplishment of something beautiful. Auschwitz is anything but beautiful. Even Bruno seems to intuitively understand this, as he recoils from touching the cold plaque.