Why the more-luminous star has larger surface area when two stars have the same temperature? If two stars have the same temperature but one is more luminous than
  the other, the more-luminous star must have a larger surface area, and
  therefore a larger radius than the dimmer star. 


The light density of the two stars is obviously different. Must the brighter star have more lights?
I'd say your problem is that you are imagining the light rays emanating out from the center of the star.  Try it again with light emanating from each square area on the surface, and you will see that being at the same temperature means each square on the surface sends out the same number of rays, but there are just more such squares.  Then you will see a lot more light rays coming from the larger star, in proportion to its surface area.