I'm confused about what the relationship between aperture size and distance from object to camera? Why does the background become blurer and blurer with the increasing of aperture?
Aperture determines the opening of the lens with which light enters a camera. Similar to our iris which expands or shrinks depending on the the amount of light present. So if you take too much light in, it makes sense that the picture would be blurry. So you can go from very sharp photos if the aperture is small to very blurry photos if the aperture is big.
Here's an answer on stack overflow. Basically, rays that enter a larger aperture can be from various directions; the ones that don't enter the camera straight-on don't focus onto the sensor exactly, so appear blurry. A smaller aperture only lets in rays that are roughly parallel, so they all pretty much focus onto the sensor at the same position.