The mishna, Shabas 14:4 (daf 111 amud 1), cites R. Shim'on as stating that "all Jews are children of kings" (and therefore can use a certain oil on their skin on Shabas).
In what sense is it the case that all Jews are children of kings? Certainly we're not literally all children of kings. Doubtless most or all of us (except for the ten lost tribes) are descendants of kings, but not via patrilineal descent; but, then, most non-Jews are doubtless descendants of kings, too. Or if it means we're like children of kings, in what sense are we so? How are we like descendants of kings? Does it mean merely that we're important, or honorable, or something, or what does it mean?
Once that's clarified — once we know what "children of kings" means — why does being children of kings effect the halacha that we can use oil on Shabas? (And other halachos, too; see e.g. Rashi to Shabas 59:2.) Why should children of kings (again, as defined in the answer to #1 above) have special halachos?