How do homogenous coordinate transformation matrices work?

The documentation of the AIR software by Roger P. Woods provides a very nice description of homogenous coordinates.

In FieldTrip, as in many other software packages that deal with anatomical and functional MRI data, we use the homogenous coordinate transformation matrix do describe the relation between voxel coordinates, where the first voxel in the volume is described as [1,1,1], and the head-coordinate system where coordinates are expressed in mm, cm or m relative to some anatomical landmarks. Following ft_read_mri the mri.transform field contains the transformation matrix. Following ft_volumerealign that matrix is updated.