What does the "Resample image" option do in Preview?
Resizing an image changes the size that it will print at without changing the number of pixels in the image. Notice if you disable resampling, "pixels" is no longer an option on the drop-down menu next to Height and Width, because resizing does not change the number of pixels, so you can't express the new size in pixel dimensions.

Resampling an image changes the number of pixels in the image. Resampling would generally be used when preparing images for the web, since you have a target PPI (see below) and need to reduce the pixel dimensions while maintaining that PPI.

One important thing to know before going further: The number of pixels in an inch, or PPI, determines the sharpness of an image. The more pixels in an inch, the sharper an image is. If you have an LCD monitor whose physical dimensions are 12" wide x 9" high, with a resolution of 1920x1080, it would have 160 PPI. A monitor with the same resolution but physical dimensions of 16" wide x 12" high would have 120 PPI and would look noticeably more grainy or blurry.

Resizing

If you make an image smaller without resampling, the resolution will increase to compensate, since when you resize, the number of pixels present does not change. 

If you make an image larger without resampling, conversely, the resolution will decrease. Your image will become blurrier because while no new pixels are being added, the pixels are spreading out to cover more space and PPI drops.

Resampling

If you make an image smaller by resampling, the number of pixels present will be reduced. This means PPI stays the same, since while you are reducing the dimensions of the image, you are reducing the number of pixels present at the same ratio.

If you make an image larger by resampling, the pixel dimensions will increase and PPI will again stay the same. In order to accomplish this, pixels have to be created where there weren't any before. Your image will again become blurrier. The program has to estimate what pixels should be added based on the existing pixels, and it won't be perfect, resulting in jagged lines and overall blurriness.