wikipedia page is pretty informative, so I do not bother to create a new one. It illustrates nicely that the length L you look through the stellar atmosphere is equal in the center and the limb of the star; however, you do not look down to material of the same temperature. And because in the disk center the material you see is hotter, this part of the Sun is brighter, too.
Now I finally come to my diagram in the beginning of this blog. It shows the brightness of the solar disk from one limb to the other. I determined this from SDO images (July 20, 2015) averaging several pictures stretching over almost five hours. The brightness is normalized to one, meaning that I just divided it by the brightness in the center of the disk. You can see the steep rise in brightness on the left limb of the disk, the maximum in the center, and the decline to the right edge. I averaged over about dozen images and the one-sigma error bars indicate how much the solar surface brightness has varied in these roughly five hours. For most points this is a change of at least several percent, so the solar surface brightness is varying all the time on a small scale.
That the disk is not really uniformly bright can be seen especially well in starspots. You can see a few in the SDO image. However, you can also see brighter regions, e.g., on the right side of the top diagram. I zoomed in to the region where a bump is visible. Here a small part of the disk is brighter than the surrounding area, and it is also quite variable because the error bars are large. We also should not forget that the Sun is rotating and features on its surface are slowly moving to the right; although this effect is very small during a couple of hours, it might contribute to the variability.
There is much more to talk about concerning this topic, but I will close for now with this: If the Sun would not have limb darkening, its brightness distribution from one limb to the other would look like the dashed magenta line in the diagram: full brightness everywhere. Not having a uniformly bright disk has severe effects on exoplanetary transits because the shape of the transit depends on how exactly the distribution looks like. I will talk about that in one of the upcoming post.