Why and how this rule works is a tricky one to explain, the best answer can be found from it's origins by a 11th century Italian mathematician Leonardo Fibonacci and his golden ratio theory (don't worry I'm not going to get too philosophical).
Many camera phones and instant cameras have the option to make a grid like this appear on the screen. The grid makes it easier to compose the image into the rule of thirds, hence the screens are always divided into blocks of 3x3.
The above two images are photographed at the same locations, but the photo on the left is composed without the rule of thirds. While the image on the right has the castle, sky line and path all placed following the rule of thirds. As you can see it makes for a more interesting image, rather than the castle shot straight in the middle.
Remember these rules are not set in stone, but before you start breaking the rules it's best you know them first.