Audio book reading is actually a type of acting, using diction and delivery to narrate a book with nothing more than one's voice. As such, producers of audio books tend to select professionals whose voice matches a particular project (though in some cases they may use other people, such as the author himself in the case of an autobiography). Becoming an audio book reader isn't easy, but it constitutes a unique form of creative expression: merging the author's voice with that of the person doing the reading.
Read books aloud, either to yourself or in circumstances which allow you to do so on a regular basis (such as a school or a children's library). Homes for the elderly often need readers, as do services for the blind. Volunteering with such organisations provides good experience for reading books professionally.
Take acting classes and work professionally as an actor if you can. That helps train your voice to deliver a narrative properly. Focus on elements such as diction, pronunciation and delivery, as well as vocal techniques for conveying character.
Network. As you develop your acting skills, make contact with talent agents, publishing editors and fellow actors who may be able to put you in touch with an audio book producer. Work to keep in touch with them, and organise their contact information in a file or somewhere else where you won't lose them.
Prepare a demo CD of your voice reading various narratives. Try to aim for as much variety as possible: different voices, different characters, different styles of books, and even different languages or accents if you can manage it. Talk to agents or publishing houses about the sort of things you should include on such a compilation.
Send your demo CD to as many helpful people as you can think of: talent agencies, small publishing houses and anyone who might have a way into the industry. Follow up after you've sent it out to make sure they've received it.
New York and Los Angeles are the centre of acting work, and while audio book readers can conceivably work from anywhere, those cities have the agencies and contacts where many audio book producers go. If you're serious about becoming an audio book reader, you should strongly consider moving to one or the other.
Patience is a virtue when applying for this kind of work. Be persistent and optimistic, but also be prepared for a long, slow climb.