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I just figured it out for the billionth time, so here it goes for everyone reading this that has typed "apply refine edge" or "refine edge doesn't $#!%ing work" into Google to find this page. It really blows me away that so many "tech savvy" guys and gals in the field can't just answer this question for everyone simply and straight forward.
STEPS:
1. Make your selection.
2. With your selection made, and the layer that contains your selection highlighted, go to your Layers panel and click the "Add Vector Mask" button. Its the one that looks like a box with a white circle cut out.
Now, you should see a mask on the layer next to your selected image, and it's inverted (black and white). And don't panic, the selection lines do go away when you do this.
3. NOW, on that layer, you can refine your edge, and when you click Apply, the changes will stay.
How hard was that to explain? I wish all the other pages I visited trying to figure it out (to no avail) had just spelled it out like that.
Happy refine edging folks!