A linear transformation maps straight lines to straight lines, that is a straight line in the $(t, x)$ coordinates maps to a straight line in the $(t', x')$ coordinates.

This is required because a body moving freely travels in a straight line, and any curvature would imply some external force was being applied. If the Lorentz transformations were non-linear then it would appear that an external force was being applied in some frames but not in others.

A linear transformation maps straight lines to straight lines, that is a straight line in the $(t, x)$ coordinates maps to a straight line in the $(t', x')$ coordinates.

This is required because a body moving freely travels in a straight line, and any curvature would imply some external force was being applied. If the Lorentz transformations were non-linear then the trajectory of the body would be a straight line in some frames but not in others. That is, it would appear that an external force was being applied in some frames but not in others.