I sat and stared at this painting for about a half-hour at Seattle Art Museum yesterday.
I think what initially hooked me in was the idea that the two circles in the background were drawn freehand, as a way of establishing the artist's skill, so that you could then go on to assume everything else about the painting was intentional.
If that is indeed the circles' purpose, then it's a way to accomplish that purpose which has a uniquely timeless quality, as if to invite a viewer looking for meanings in this painting to favor those which likewise have timeless qualities, in this case given that the painting is a self-portrait presumably ones about the human condition.
In that light, what I couldn't turn away from was the powerful impression this painting conveyed that its subject had transcended aging - that through mastery of his profession and the resulting creation of works which would bring benefit to the world in perpetuity, death was no longer anything for him to fear.
And that itself seemed to stand in sharp contrast to many other artworks of this era and earlier, in which whether or not to fear death was bound up in religion and the whims of gods and goddesses rather than in anything which was fundamentally under your control. So, what I finally came to was that this self-portrait seems so powerful because it revels in one of the ultimate benefits of the Renaissance and one of the key advances of our species.