The basic idea is this: your frame is broken down into nine imaginary but equal parts. By placing your subject in the areas where the lines intersect, you create a natural balance in your image, making it easier for your viewers to take in. According to research, humans are hardwired to view images in certain ways, and our eyes naturally focus on these intersecting points, rather than the center of the frame.
I placed lines within the frame of this picture to give you an idea of what I mean.
Creating this frame in your mind while you are shooting can be difficult for beginners, but with a little practice, this skill can become second nature!
As you get better at composing photos, you will come to realize that this 'rule' is a great tool for balancing the elements within your frame, but is to be used as more of a guideline than something etched in stone.
This is the case with most rules of photography, and rules were meant to be broken, right? Some of the greatest photos taken both adhere to this rule and also break it! If you can create a more visually pleasing photo by bending ignoring this rule, or any other rule of photography, take aim and shoot!