"So what does this do to the speed of light being a constant and an absolute limit?"
NOthing at all.
Nothing can move through space at >c, but space itself can expand at any speed. In fact there are many distant galaxies that will will never see because they are already receding from us faster than the speed of light. The galaxies can be not moving, or moving relatively slowly in relation to the local space the exist in, but the vast amount of space between them and us is expanding.
That's part of it. Sort of. The inflationary phase was over long ago. The expansion of the universe, a completely different mechanism, continues even now, and may be speeding up.
Or so goes the theory...yet the space between us and the Sun isn't expanding.
Why is some space expanding, but not other space?!