Why Maxwell equations are not invariant under Galilean transformations, but invariant under Lorentz transformations? What is the deep physical meaning behind it?
The simplest answer is that Galilean transformations do not preserve the invariance of light's speed but Lorentz transformations do. Are you looking for something deeper?
It tells you that the phenomena of electromagnetism are inherently relativistic and unlike mechanics there is no "Newtonian" low-velocity non-relativistic limit. Hence the equations have Lorentz structure built into them.
This is because electromagnetic radiation is a manifestation of some properties of the spacetime, consequently it has to be invariant under those transformations which preserves the spacetime interval invariant.