The factor of Z in the denominator takes into account the protons in the nucleus. Due to its attractive force on an electron, each additional proton will "pull" the electron into a tighter orbit. Thus, the radius of the first energy level of carbon (with its 6-proton force) will actually be considerably smaller than the radius of the first energy level of Hydrogen (with its 1-proton force.)
The reason for the factor of (Z-2), for n=2, is more complex. As the second electron energy level begins to fill, the force from the protons that these n=2 electrons "feel" is decreased slightly, by the n=1 electrons in between. That is, the electrons in n=2 are partially shielded from the nucleus by the electrons in n=1. (And the electrons in n=3 are shielded by both the n=1 and n=2 electrons, and so on.) By replacing the factor of Z with (Z-2), we estimate that the radius of n=2 is influenced by a charge of +4 (not +6), since the two electrons in n=1 cancel out two of the proton charges. Again, consider this equation an approximation at best.