In this video on Khan Academy, Sal remarks that 'all atoms envy the noble gases, which are the most stable because they have eight electrons in their outer shell'.
I can't think of much of a reason that the number of electrons in the outer shell would affect the stability of the electron configuration. Additionally, I read that more electrons in the outer shell slightly increases the atomic radius. A larger atomic radius usually indicates a less stable electron configuration.
However, perhaps I've been thinking about this the wrong way. I've been wondering whether some property obtains when there are eight electrons in the outer shell that causes the configuration to be more stable. Perhaps, I ought to think about thus: atoms tend toward having eight electrons in their outer shell (i.e. eight electrons in the outer shell coincides with configuration stability, but doesn't cause it). The fact that the unscreened Coulomb force increases from left to right along a period is what explains why atoms tend toward having eight electrons in the outer shell: The atoms near the left of a period exhibit relatively little unscreened Coulomb force, and so are prone to giving up electrons and, so are prone to giving up the few electrons that constitute their valence shell. Whereas, those atoms near the right of the period exhibit much Coulomb force, relatively. Accordingly, those atoms are prone to attracting more electrons, and thereby prone to filling their valence shells.
Why does the stability of the electron configuration increase when the outer electron shell is full?
NOTE: I have reviewed the question Can an atom have more than 8 valence electrons? If not, why is 8 the limit? That question asks about the maximum number of valence electrons. This question asks why eight electrons in the valence shell are more stable than 1, 2, 3, ... 6, or 7 electrons in the valence shell.
