For most of human evolution -- 2.5 million years -- we ate "caveman" or Stone-Age style: by hunting, fishing, and gathering. It's only in very recent times -- about 10,000 years ago (a very small fraction of our entire evolution) that we began eating other foods like dairy, grains, beans and other crops that are based on farming. Paleo enthusiasts believe our bodies have not fully evolved to handle the agriculture-based diet, so we are better off returning to a hunter-gatherer diet.
However, other evidence suggests that humans may have been eating grains and tubers for 100,000 years, long enough to fully evolve the ability to tolerate them. Other experts point out that hunting isn't always a successful activity; to survive, our early ancestors would have had to eat at least some non-meat foods, namely plants, to survive.
Regardless, the idea of Paleo is that you shouldn't be afraid to eat a lot of meat -- even the majority of your calories as meat if you want to. There have been no conclusive studies that show eating meat causes the health ailments so many claim. Our bodies are quite capable -- even superbly capable -- of processing meat (remember those 2.5 million years of evolution?).
The diet war between plant-based and meat-based advocates rages on endlessly. In reality, the debate is pointless because BOTH diets are focused on whole, real, nutritious foods. Really, it's ideal to combine the two diets. A high protein, high plant diet is the ideal weight loss diet (see Magic Meals). The real "bad guys" are the calorie dense "junk foods": refined carbohydrates (sugar, white flour) often combined with unhealthy fats, that are addictive, hard to stop eating, and contain little nutritional value.
A Paleo diet is a great starting point for diet and nutrition beginners because the "caveman" theme is so easy to remember. Except for simple cooking methods, Paleolithic man ate food that was very close to a natural state -- food that was gathered in the wild and consumed with minimal processing.
They didn't have the machines or the time and energy to process or manufacture food (such as the flour and sugar so common in our diets today). They hunted and gathered. They ate plenty of meat and fish, they gathered nuts, seeds, berries and other fruit, and when meat or fish wasn't available they gathered naturally occurring plants, berries, seeds and tubers (roots).
Paleo is basically a high protein, high-fiber, low carb diet. Strict Paleo enthusiasts also rule out any kind of dairy (cheese, milk, etc).