I guess what I really want to know is what I am getting into in repairing it. I wonder why this strange gauze was used, and if the leak goes down past the floor.

Could be condensation and an attempt to limit it.

So has no one seen this type of repair before?

I am planning to remove it with a razor blade and see what is inside so that i can attempt repair, just want to make sure I know what I'm getting into.

I can't quite figure out why someone would repair in this way.

No idea. It will be great if you posted up your findings though, for future searchers!

As brickeyee said, are you sure there is actually a leak and it's not just insulation to (unsuccessfully) stop condensation?