What is the meaning of six of one half a dozen of the other?

"Half a dozen = 6. So "six" and "half a dozen" are two ways of saying the same thing. The expression means that there is no important difference between the alternatives, or the differences offset one another so the net result is the same.

For example, I say to my husband, "Should I take Highway 101 or Highway 280?" and he replies, "It's six of one and a half dozen of the other." He means that I'll get there in about the same amount of time whether I take one road or the other."

The phrase, "Six to one, half a dozen to the other." is a UK variation of the phrase.

This person's answer says much but conveys little.

"Six of one, half a dozen of the other" is a reply to a question that solicits an evaluation between two choices. The person giving the reply is effectively saying "there is no difference between these two choices".