Exposition is generally the background information that helps give the story a frame in which to be read and understood. In the case of "The Destructors" this frame can be attributed to the setting, the historical background and the social situation of the young protagonists.
The story is set in England, after WWII and as the Depression looms. It was a time of physical, mental and financial recovery from a war with an unsettling, pessimistic view of the future.
The boys in the story call themselves the Wormsley gang and are reminiscent of the teddy boys from the era. The teddy boys were packs of juvenile deliquents who wreaked unexplained havoc where they went - thus the title of the story. These boys frequent areas of town destroyed by bombs and look for ways to pass the time and have fun with their devastation.
Their target is a man's home. Their senseless destruction and terrorization of the man and his home are indicative of the attitudes of their generation and the harsh reality of their world.