Moving a Reef Tank

Curious about your answer. If a tank can support the water and all of the
rock and sand when it's displayed, why can it not when being moved.
Another forum recommended that I move it with just the sand in it, and
enough water to cover said sand. I was able to do that with no problem.

So you assume that everyone's tank can handle it. Dangerous assumption!

Also, what is the purpose of keeping the original water.

This is might get a little too complicated for you. More original water
means more bio load when you set up.

I was able to transport about half of the water and replaced about half
when I got here. But it seems that it's just like any other water change.
I was under the impression that most of the bacteria that is needed is in
the rock/sand/other filtration, very little of it in the water.

Bad assumption. This is why your bioload is not as high as mine with no
water changes.