How exactly is an LCD damaged by DC current?

A LCD segment consists of a cell filled with liquid with very thin transparent electrodes on either side.

If a DC voltage is applied to this metal is transferred from the anode to the cathode resulting in the anode getting thinner. Eventually the anode will fail or its resistance will increase reducing its effectiveness.

Excessive current will be caused by metallic whiskers growing between the electrodes by the same mechanism effectively shorting the electrodes. Imagine that you have a high point on one surface the electric field is stronger at the high point so it attracts more metal ions making it higher making the field stronger......

An article on Focus LCD's article Segment LDC displays avoiding excessive DC component states this:

Results of excessive DC component on a segmented LCD Extended operation of an excessive DC component on a static display, or segment LCD, can damage the display by electro-plating the electrode contact. Noticeable results of a damaged display can include a blurry segment (decrease in contrast) and the segments can become burned in.