The formula you mention for $n_i$ is empirically supported (i.e., it is an empirical fit to experimental data for a solid material) and is indeed an average in a cubic centimeter of material. $E_g$ is the band gap energy for the material of interest and B is an empirically determined constant. Thus, normally, with access to a table of such constants and band gap energies, you can determine the intrinsic carrier concentration for a given temperature of interest.
$n_i$ plays a significant role in the design of semiconductors devices because of the relationship, $n_i^2 = np$ (law of mass action) which holds in doped and undoped material. This is important for, among other reasons, doping profiles and for understanding how a material acts a different temepratures. However, given that most semiconductor components are much much smaller than a $cm^3$ in volume (state of the art transistors have a linewidth of $14-22 nm$ and the junction depth is a few hundred angstrom), you can see that the of scale for $n_i$ is rather meaningless to an individual component and an average would be necessary anyway.
One last point, the reason why the equilibrium relation, $n_i^2 = np$, holds true is because in a crystal structure, the generation of hole-electron pairs and recombination is constantly occurring in the crystal because of thermal agitation in the material. Take the equation for $n_i$ and let the temperature go to absolute zero, you find that the intrinsic carrier concentration also becomes 0. The meaning of the temperature increasing is an increase in kinetic energy/vibration of atoms or equivalently an average increase in the internal energy in the crystal. This is because temperature is a function of the internal energy (U), $T = (\frac{\partial{S}}{\partial{U}})^{-1} = (\frac{Nk_b}{U})^{-1}$. So, there are more opportunities for bonds in crystal to break and reform.