In my tiny, universally insignificant, mind I think an early post of "it's like blowing up a balloon!" makes the most sense.
You take an uninflated ballon and with a Sharpie you put two dots on it right next to each other. Traveling at the speed of light it takes an instant to move between these two points.
Now you blow up the balloon. As the air you put in it expands the balloon the two points on it now move further and further apart from one another meaning that now traveling at C between these two points takes longer. The two points haven't really moved the space between them has expanded.
Now, let us say that at the begining these two points were a "light year" apart in physical distance. When you expanded the balloon you had to also expand your "concept" of what a light year is. You cannot change one without changing the other. So, NOW those two points are still a "light year apart" but now that light year is a greater distance. Since light moves at a constant speed and cannot be changed (unless you change the medium it is traveling through, through a medium c is a constant) it will now take light longer to move between those two points.
As someone above said, the universe's continuum is expanding and dragging the matter of the universe along with it.
Stop thinking of the universe being a vast, empty, space and the galaxies within it are moving apart from one another like the plasma of a fire-explosion and thing of, well, a baloon.