What is the purpose of an etched glass? What does etching on the bottom of a beer glass do? What purpose does an etched glass serve?
The etching provides a nucleation point. To quote a couple places:

A nucleation point on a beer glass refers to an etched mark or pattern on the bottom of the inside of a beer glass.  The etching is called a nucleation point (or a widget in the UK) and helps the release of carbonation and can create a steady stream of bubble emanating from the etched portion of the glass. This works by CO2 releasing (dissolving into gas) when it comes in contact with the rough surface of the nucleation points.  This is not a mere marketing ploy and a nucleation point does increase the amount of bubbles released when compared side by side with a non- nucleated beer glass.

And

A laser etched nucleation site within the glass maintains flavor release during the drinking experience. Samuel Adams Unveils the Samuel Adams Boston Lager Pint Glass.