Respect for the Office

I do not understand this concept, and I'm hoping some clever readers might be able to help out.

Over and over, we are told that even if we disagree with the presidents' policies, we should "respect the office". What does that mean, exactly?

Is the President, by virtue of attaining the office, somehow in possession of superior physical or mental attributes, and thus should gain extra respect on account of physical or mental superiority to the rest of the citizens?

Or maybe the Office of President requires great concentration, and puts many heavy burdens on the man who holds it, and the respect asks that I give the President a wide berth so as not to disturb or distract from the weighty tasks?

Or is the respect one of fear, because the president, by virtue of the office, is capable of wielding great power and popping me in the slammer (or worse) at whim?

Perhaps, the respect for the office comes from the idea that the "Leader of the Free World" is a great humanitarian and uses the office to do wonderful things for humanity?

Is there another angle I'm missing? The second explanation is the most logical, but I never see that approach demonstrated in the application of the phrase "respect for the office". When Pittsburgh Steeler James Harrison turned down a social invitation to the White House, he was quite fulfilling the second explanation, and not taking the President away from his weighty and critical tasks, yet the Super Bowl winner was widely panned in the media for 'not respecting the office'. So, clearly not an application of the second, and most logical, interpretation.

What does it mean, then, "respect for the office"? Where does that respect come from, why is it deserved, and what does it demand that is different from respect for any other American?