Local contrast enhancement attempts to increase the appearance of large-scale light-dark transitions, similar to how sharpening with an "unsharp mask" increases the appearance of small-scale edges. Good local contrast gives an image its "pop" and creates a three-dimensional effect — mimicking the look naturally created by high-end camera lenses. Local contrast enhancement is also useful for minimizing the effect of haze, lens flare, or the dull look created by taking a photograph through a dirty window.
The trick with local contrast enhancement is that it increases "local" contrast in smaller regions, while at the same time preventing an increase in "global" contrast — thereby protecting large-scale shadow/highlight detail. It achieves this feat by making some pixels in the histogram cross over each other, which is not possible when enhancing contrast using levels or curves.
Local contrast enhancement works similarly to sharpening with an unsharp mask, however the mask is instead created using an image with a greater blur distance. This creates a local contrast mask which maps larger-scale transitions than the small-scale edges which are mapped when sharpening an image.