Are you really asking why the speed of light is invariant regardless of the observer's relative motion? I will try to answer that, although I'm really rusty on Maxwell.
It was commonly thought that there was an "aether" through which light propogated, just like sound waves in matter.
If so, since the earth is moving through space at pretty high velocity, if you measure the speed of light along tubes oriented at right angles, you should see a difference.
That was the Michaelson-Morley experiment, and oddly enough, there was no difference.
It appears as long as you're moving steadily in a straight line, no matter where or how fast, if you measure the speed of light it comes out the same.
In order to explain this puzzle, a new theory was born, Special Relativity.
It says our clocks and our measuring sticks (the things we measure speed with) start acting funny when we go fast with respect to someone else.
Here's how it works.
Suppose you build a clock by having two parallel mirrors in a vacuum one half meter apart, and you bounce a small pulse of light vertically between the mirrors.
You count the bounces. 299792458 round trips takes one second, because that's how fast light goes.
Now you mount this clock on a railroad car and look at it as it travels past you.
Since the light is now traveling in a slanted direction, because the whole clock is moving, the light pulse has to travel further between bounces, which will take longer from your perspective.
So from your perspective the clock on the train is running slower, but from the perspective of the person on the train, it appears to be going the same speed because that's the only clock he's got.
In fact, if you the supposedly stationary person also had a clock, the person on the train would say that your's appears to be running slower.
So, if the Michaelson-Morley experiment is correct, this is what you and the person on the train should observe.
Well, experiment bears it out. That's what actually is seen. When very precise clocks are compared, and one of them is moving in a straight line at a constant high speed relative to the other, it does run slow.
The faster it's moving, the slower it runs.
In theory, if it could go at the speed of light, it would stop.
Also, measuring rods aligned in the direction of motion are shortened, for similar reasons.
So if you're trying to explain why the speed of light is constant regardless of the observer's motion, the answer is, we don't really know, but it appears to be, and the implications are that clocks and measuring rods should do funny things.
And in fact, they do. This has deep implications, for better and worse, including nuclear energy and warfare.