The ellipse is the polar set of middle perpendiculars on BQ, if Q runs on a circle around A. The point P on the ellipse, clearly satisfies AP+PB=AQ. It is less obvious to see, that the middle perpendiculars are indeed tangents to the ellipse, i.e., the ellipse lies on one side of them.

If you look at this construction, you see that the ellipse is the track of all points with equal distance to a given point and to a circle.