Why would things like this happen?
I think it's due to how densely we sample the texture. For example, on the far left case of magnification, we're essentially zooming in on the texture, such one patch of the texture takes up many pixels on the actual screen. The far right case of minification is zooming out, so there's less precision than in the texture.
The pixels can also become parallelograms rather than squares because of anisotropic transformations, where the two dimensions (x and y) don't have the same scaling and are not parallel to the texture axis.