No matter the frame light is in, it always moves in a straight line in that frame. Why is that? It doesn't seem like something to me that should necessarily be true. If some one runs forward and sends something perpendicular to their motion, why would it continue to move forward with them-it doesn't carry the source's velocity.
Everything moves in geodesics if not acted on by a force other than gravity -- this is an axiom of general relativity (the geodesic equation). Geodesics are straight lines in the absence of gravity -- this is part of the other axiom of general relativity (the Einstein-Hilbert action, or the EFE or whatever).
Light doesn't interact much with everything, except quite weakly with gravity, and with some miscellaneous scattering patterns, like those which allow you to actually see things, but those are quite pointy (reflection, refraction, etc. -- the paths are pointy as long as the scattering boundary is sharp), so you still see a bunch of straight lines.
It is an axiom of general relativity. A basic assumption. A more accurate statement would be that light always travels along a spacetime geodesic. The classical addition of speeds in newtonian meachanics doesn't work inrelativity (light is the most extreme case). That is the source of all weird stuff in relativity
The straightness of light rays follows from constructive interference of waves that approximately follow the shortest path (or more generally an extremal path).
Around the extremal path the derivative of the path length wrt variations of the path is 0. Therefore approximately extremal paths all have the same length and thus the same phase making waves following them interfere constructively. Paths around non-extremal path interfere destructively and waves following them cancel out. This explanation applies classically to the EM field and Maxwell equation, or quantum mechanically to field of amplitudes. This is also the explanation of Huygen's principle.
Feynman gives a detailed explanation of this in QED, the strange theory of light and matter as well as in his lecture notes and other places.
The precise statement should be that light in free space moves in straight lines and constant velocity wrt inertial reference frames. It does not move in straight lines or constant velocity in accelerating reference frames or in non homogeneous media.
The fact that the speed of light is independent of the observer's velocity is a relativistic effect. Formally it is a postulate of special relativity.
In general relativity light follows geodesics wrt the spacetime metric.