During the resting period, starch molecules in the flour are absorbing the liquid in the batter. This causes them to swell and gives the batter a thicker, more viscous consistency. Any gluten formed during the mixing of the batter is also getting time to relax, and air bubbles are slowly working their way out.
This resting step ensures a thin and uniform structure to the finished baked good, as with the hollow shells of popovers or the paper-thin quality of crêpes. And because the gluten had time to relax, the texture is incredibly delicate instead of chewy.
If you're in a rush, you can shorten or skip the resting step, though the texture won't be quite the same.
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Related: Kitchen Science: Demystifying Gluten
(Image: Sara Kate for the Kitchn)