Circles Of Confusion
Explaining exactly why changing the size of the aperture alters the depth of field is going to get a little bit technical, but I've always found that it's a lot easier to understand and remember something if you know how it works. Don't worry, I'm not going to go into the maths here, but I am going to inflict a couple of diagrams on you.
This is a highly simplified diagram of the arrangement of lens, aperture and sensor inside your camera. In this first diagram, our subjects are the three spots, red, green and blue, which are at different distances from the camera. The camera lens is focused on the green spot, which means that light from the green spot passes through the aperture and the lens and is focused on the CCD. Light from the red and blue spots also passes through the aperture and lens, but light from the red spot focuses a short distance in front of the CCD, while light from the blue spot would focus a short distance behind it. The light from these other spots still hits the CCD, but due to light scattering it is unfocused and spread over a wide area.
What this means is that the red and blue spots will appear as large blurred spots on the final image, while the green spot will be sharp and in focus. The size of the blurred area of the red and blue spots is called the circle of confusion.
In this diagram we have the same setup, and the coloured spots are the same distance from the lens, but this time the aperture has been reduced to just a small hole. Again the lens is focused on the green spot, and the red and blue spots are out of focus. However the narrow aperture restricts the light scattering and the relative angles of the light paths, and as a result the "circles of confusion" are much smaller. This makes the red and blue spots in the final image appear much sharper. They are still out of focus, but the effect is not so noticeable. To make circles of confusion as large as in the first image, the red and blue spots would have to be much further away from the green one.
How aperture works.
Circles Of Confusion