Without going too deep into the symbolism, and to piggyback on dfreybur's post, before I came to the Craft, my grandfather passed away. He was Mason of over 60 years and had never spoke to me about it. However, from his death, I saw his Masonic funeral and also came across a Masonic publication when going through his things. In it, a brother wrote an article on how to be a better Mason. I, unfortunately, no longer have a copy, but part of it talked about striving for fairness in all of our dealings. An example would be paying at a cash register and having the wrong amount of change returned. The moral was that just as we should not allow the cashier to give us back less than we are owed, nor should we allow them to give us more.
The square measures a perfect angle and ensures that it is no more or less than 90 degree. It is not biased or swayed by personal motivations. As such, the WM, while ruling over his lodge, ought to be sure to square his actions in his dealings with the lodge as a whole and to see that he is ruling virtuously and is neither too generous nor too frugal regardless of his personal feelings. As the saying goes, 'with great power comes great responsibility' and the square reminds the Master to rule impartially.
Now...there might be additional lessons to be learned that I am unfamiliar with as I am not a PM, but that is my take.