If you're mathematically inclined, look at: http://en.wikipedia.org/wiki/Transformation_matrix

For a less technical description: The pixel values in the transformed image are generated as a function of the pixel values in the input image. Once you have the transformation function figured out, the rest is basic algebra.

If you want to get an idea of what these functions look like, flip to the "Examples in 2D graphics" section in the above Wikipedia link. There's examples there for rotation, resizing, shearing, and others.