What is the significance of Maxwell's equations being invariant under the Lorentz transformation? It seems maybe this has something to do with an overarching notion in Physics that things that are symmetrical look nice but I need to be educated on a deeper level about how this impacts the outcome of whether a theory is right or wrong. I have taken an undergraduate EM course and recently started reading special relativity on my own. Nowhere in the course did it mention transformations. So I'm still trying to wrap my head around the popular statement that Maxwell's equations are invariant under the Lorentz transformation. What does this mean and are there any examples in daily life that this can be applied to? More generally, why do we like transformations that don't change the physical laws? Is it a problem if a transformation does change the laws? Does the fact that the equations are invariant under Lorentz make Lorentz a "correct" transformation in the sense that makes it wrong to say that if someone is moving at 100 and I'm moving at 90 in the same direction then he's moving at 10? Does the Lorentz transformation have any other advantages over the Galilean other than the fact that it preserves the speed of light? Please do not judge the phrasing of my questions too harshly. I hope you get  the gist of the matter. I just mean to know why we said that phrase in the first place and how it makes physics tick.
 A: It's not really a matter of symmetry groups of equations. A simple model of waves on water is $\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0$, and this is invariant under Lorentz transformations with speed of light $1$ (for the same reason $dt^2-dx^2$ is invariant under Lorentz transformations with speed of light $1$). The invariance of the simple wave equation didn't have any significance for physics.
The idea, thinking as a physicist in the year 1900 or earlier, is that either Galilean invariance is wrong or Maxwell's equations are wrong.
Possibility one: there is a privileged reference frame in which Maxwell's equations hold, possibly complicated by a "luminiferous ether" which might be "dragged along" by Earth like a fluid. The fundamental laws would obey Galilean invariance, but Maxwell's equations hold in the frame in which the "luminiferous ether" is at rest. No one ever got anywhere or did any useful physics with this approach.
Possibility two: Maxwell's equations hold in all inertial reference frames, and Galilean relativity is simply wrong. This seemed untenable to all but Einstein, because it implied no notion of simultaneity can exist, that time passes differently for different inertial observers, and all other weird special relativistic phenomena. However, it has provided countless experimentally confirmed predictions; it's how nature works.
A: 
More generally, why do we like transformations that don't change the physical laws? 

It's not so much that, or at least it wasn't at that time.
The problem was that we knew that normal Newtonian mechanics was invariant via the Galilean transform. That means that non-accelerated frames have similar physics. However, there was no analog of this in Maxwell's equations.
If you throw a ball from A to B in an inertial frame then the results are the same if that frame is at rest or moving. And we're happy with this, because the person that's tossing the ball and the person that's catching it are still at rest relative to each other, which just "makes sense" (although they didn't always think that of course). The results are not the same if I throw the ball between the frames, because then there's relative motion. In that case we have a transformation we apply, the relative speeds of the frames cause additional terms to appear.
So now here's the problem - what if instead of a ball, I toss some radio waves? At first glance it might appear that the same thing should apply, because the transmitter is moving in the same frame as the receiver, so the waves naturally have a relationship to each other at either end of the experiment. But in that case, we would also naturally expect that waves moving between frames would not maintain their relationship. And that's the rub...
If this is true, it would be immediately visible - the color of planets would change as they moved around the solar system, for instance. Look as they might, no one was able to find such a thing. It appeared that light was not following the Galilean transform. Quite the opposite, every experiment showed no transform at all, all light was emitted in a single universal frame that had nothing to do with matter.
And that's bad. Because we also knew that light did interact with materials - you can shine light on a piece of metal to heat it up, for instance. Well if light is in a fixed frame, then what happens when I put a light bulb on a cart beside a sheet of metal? Wouldn't it heat up faster or slower depending on which way I push it? And if the fuel for the light is on the cart, then don't I create free energy if I push it in the right direction?
That would be bad. Like universe exploding bad.
So it's not that people "like transformations that don't change the physical laws", it's that we already had one transform. Either light also had some sort of transform that made things work out again, or we had to throw everything in the trash and start over.
And thus Lorentz. His transform magically makes light work in Galilean frames, which is the very reason he did it.
A: Einstein may not have developed the theory of Special Relativity if the Maxwell Equations are not invariant, as they led him to believe that the notion of absolute space and absolute time (everybody everywhere will agree on the same measurement) was wrong. He had to decide who was right, Newton or Maxwell.
It is a problem if physical laws are not invariant, a silly  example is if we sent a spaceprobe to Mars, Pluto or Alpha Centuri and then found out the physical laws were different there.
Also, our judgements on astronomical distances, the luminosity of stars and, basically the validity of physics as an accurate description of the universe depends on the invariance of physical laws. We may learn a lot if we find that the laws are subject to change, but we have not done so to date.

Is it a problem if a transformation does change the laws?

Yes it is, because the laws must be invariant in order for them to apply to any region of spacetime. That's why, if you study SR and GR, the equations describing physical laws are written in co-ordinate free notation. 
A: Maxwell's equations and Einstein's relativity are totally unrelated. Some relation existed when Maxwell's theory was an ether theory and accordingly assumed that the speed of light is independent of the speed of the light source (Einstein's 1905 second postulate). Nowadays the statement "Maxwell's equations are invariant under the Lorentz transformation" is not even wrong.
