Chronos said:
Inflation is the short answer. The very early universe expanded at superluminal velocity.
I really dislike this terminology. Expansion isn't a velocity. It's a rate (or velocity per distance, whichever you prefer).
Any amount of expansion is "superluminal" at large enough distances.
What separated out inflation was that it was a period of exceptionally fast exponential expansion.
Anyway, the claim being made in that book is a somewhat different claim, one that inflation does not answer. The idea is simply this: if our universe is a thermal fluctuation, it is a pretty darned big thermal fluctuation (the entropy of the early universe was absurdly small compared to the current entropy, or the eventual entropy of our region of space-time). One early concept was that this was just a natural thermal fluctuation: if you wait long enough, thermal fluctuations of all sizes will occur. So eventually you will get a region with tremendously minuscule entropy, like our own had.
However, it's much easier to make a small fluctuation than a large one. Small fluctuations happen tremendously more frequently than large ones. Instead of making a whole universe, it's a much tinier fluctuation to just make a single galaxy instead. Heck, it's an even smaller fluctuation to make a single brain. A brain that has memories of a long existence, and even has some sensory inputs that make it think there is something around it. This brain immediately dissolves back into the thermal fluctuation from whence it came, but it thinks it exists, and observes a reality.
Fortunately, we can be pretty sure that we are not such brains: nearly all of these "Boltzmann Brains" would observe a horribly disordered, chaotic universe. In order to observe an orderly universe, in order for our past experience to have a chance of predicting future outcomes, it only really makes sense that we are actually real, that our early universe actually had extremely low entropy.
But then this brings us right back to the beginning:
why did our early universe have such low entropy? The discussion above proves that it wasn't simply a rare thermal fluctuation. But then what was it? Well, there are a number of ideas. Sean Carroll somewhat recently wrote a popular science article on the subject:
http://www.scientificamerican.com/article.cfm?id=the-cosmic-origins-of-times-arrow