How do planes keep from hitting each other at such events?

The midair collision at the Wings Over Dallas air show was an exceptionally rare instance. The movements of planes at such events are closely controlled and monitored by air traffic controllers. In fact, the main question that investigators into the Dallas tragedy will be trying to answer is, "How did the collision happen in the first place?" And the answer to that question might be far more involved than most people would imagine, as they will look into the mechanical history of the planes, the pilots' history and condition, pre-flight briefing, ATC communications, operating rules used by the owners of the planes and the organization that put on the air show, and the past performance of the aircraft involved at previous air shows, among many other potential sources of information. That's why such investigations take a long time.