Galilean transformations assume that you can instantly synchronize all clocks regardless how far they are (so, they assume that speed of fastest available communicator -light- is infinity). If it is less than infinity, then "in reality" there is no way to have the same (absolute) time everywhere (and the same geometry of space, by the way) - it all begin to depend on your position (versus object) and on you state of motion - so you have relativity (Lorents transformation between coordinates and times of the same event but seen in different frames)- versus Galilean absolute space and time frame for all observers regardless their state of motion.

Because physical laws depend only on RELATIVE position and motion of observer and object but not on their ABSOLUTE position/motion in space(like versus some aether), Galilean transformations are incorrect and Lorents transformations have to be used. Thus by the way, the name "relativity theory" - only RELATIVE position and RELATIVE motion matters for all observable phenomena.