Great book! It's one of my favorite books of all time. In fact, I refuse to see the movie and let it be ruined.
The battle room is where all of the kids (students) put into practice the battle tactics that they are learning in the classroom setting. Students are divided into groups, which the book refers to as armies. Each army enters the battle room from a different side. The battle room has zero gravity, because the students must learn to fight in a three dimensional space. That is how space battles will happen against the buggers, so that is how the cadets train. Each "soldier" is equipped with a laser rifle of sorts, and hits are registered on a special suit that locks into position as specific body parts are hit.
The students learn battle strategies in the battle room. More than that though, they learn teamwork, chain of command, and creative thinking skills.