Lessons in warfare are only part of the military benefits; political motivations and gain were an equally important part of this proposal. Though he doesn't go into extensive detail, Reiffel does reference the positive position to be gained by the nation that could detonate a nuke in space, a feat he lauds as "a demonstration of advanced technological capability." It would certainly show the Soviets that the R-7 rocket that had launched the innocuous Sputnik into orbit paled in comparison to America's most advanced unmanned missiles.
Of course, it wouldn't just be a win for the United States; if the world weren't prepared for something as dramatic as a nuclear explosion in space the effects could be extremely negative and devastating for America's popularity abroad.
Unfortunately, the meat of the military side of Reiffel's proposal is missing. The details are in the second chapter, which directs the reader to Volume II of the report, which is unavailable to the public. Volume I is all about the science and makes no specific mention of the type or size of nuclear explosives Reiffel proposes detonating near the moon.