Other contributors have said "How does hyperventilation prolong the time you can hold your breath?" is the same question as "Why should you be able to hold your breath longer after hyperventilation?". If you believe that these are not asking the same thing and should be answered differently, click here.
Why should you be able to hold your breath longer after hyperventilation?
Answer:
Because hyperventilation clears out more of the carbon dioxide out of your system, and it's the build-up of CO that controls your breathing reflex.
Do note that hyperventilating is a Bad Idea, particularly when diving. Lack of oxygen is what knocks you unconscious, but it's the build up of CO which forces you to breathe. If CO build up is retarded you can lose consciousness before the breathing reflex has forced you to the surface.
It's so common that it's even got a name: shallow water drowning
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