Rather than asking why we should impose this invariance, I think it would make more sense to ask why we should relax it. The Lagrangian is a relativistic scalar, which means that it has to be invariant under any coordinate transformation. An infinitesimal change of coordinates is just one type of coordinate transformation.
The physical interpretation is that coordinates don't in general have a physical interpretation. Coordinates are just names. A change of coordinates is just a renaming of points in spacetime. If we're going to derive equations of motion from a Lagrangian, we want those equations of motion to be true regardless of how we rename points in spacetime.