The underlying reasoning makes sense to me - homo sapiens have existed for the 200,000 years (and our various ancestors for millions of years before that), and for the first 190,000 years we lived on and evolved with hunter-gather diets.
It wasn't until 10,000 years ago that agriculture was invented and our nutritional intake, lifestyle, culture, economics, and everything else changed with it. But physiologically we haven't fully overcome millions of years of adaptation to the hunter-gatherer diet.
Paleo is about constraining your diet to something as close to the hunter-gatherer diet as possible - no processed sugars or grains, but rather meat/fish/poultry, nuts, berries, roots, vegetables that don't require agriculture to obtain, some fruits, etc.
A few benefits off the top of my head - I'm less hungry, more satisfied, and have more sustained energy throughout the day, with less food. I feel 'cleaner' and healthier, and have almost no gas (the biggest revelation). And I've never had a terribly unhealthy junkfood diet, but Paleo is still a noticeable improvement.
You can also cheat and go 70%-80% Paleo, instead of strictly 100%, and still derive some benefits (enough to sell you on it), although 100% Paleo is exponentially better.
There's also no crazy initiation period like there is with Atkins and South Beach, you just go read a book about what constitutes Paleo food, and start integrating it into your diet and replacing non-paleo foods at your own pace.
I highly recommend checking it out:
http://en.wikipedia.org/wiki/Paleolithic_diet