The "Pushing" part really refers to the development part of the process. As does the "Pulling" part.
If you want to increase the contrast of your negatives, you Push development.
If you want to decrease the contrast of your negatives, you Pull development.
When are you most frequently likely to want to increase the contrast of your negatives? When the light is so low you are forced to give the film less light than normal - i.e. under-expose it.
Most people accomplish that under-exposure by using their meter set at one, two or more stops higher. Than they choose the Push development that corresponds to the amount of stops they under-exposed the film.
A Push development doesn't really increase the density of the parts of the negative that correspond to the darkest parts of the scene. What it does is increase the contrast of those parts of the negatives that are in the mid-tones or shadows that are close to the mid-tones. So they print better. At the same time, a Push development increases the contrast of the highlights, which can make them harder to print.