The expression you are describing is equation (6) from your first link: $$R_j=\frac{3\hbar c\ln(10)1000}{16\pi^2N_A}\int_\text{band j}\frac{\Delta\epsilon}{\omega}d\omega\tag{1}$$ which defines the rotatory strength $R_j$ of a band $j$ as the differential absorption coefficient integrated over that band, with the units changed via a prefactor containing the reduced Planck constant ($\hbar$), the speed of light ($c$), and Avogadro's number ($N_A$). The expression is the same for oscillatory strength, except it integrates just over the absorption coefficient, not the differential.
The prefactor has units of $\pu{g*cm^3*mol*s^-2}$ ($\hbar$ has units $\pu{g*cm^2*s^-1}$, $c$ has units $\pu{cm*s^-1}$, and Avogdaro's number is $\pu{mol^-1}$). Due to dividing and then integrating by $\omega$, the units of the integrated intensity are just those of $\Delta\epsilon$ ($\pu{L*mol^-1cm^-1}=\pu{cm^2*mol^-1}$). Combining these, we get units of $\pu{g*cm^5*s^-2}$, which doesn't look very close to the desired result until you realize that the ESU $\pu{statcoulomb}$ is equivalent to $\pu{g^{1/2}*cm^{3/2}*s^{-1}}$. Subbing this into the prior expression, we obtain $\pu{statcoulomb^2*cm^2}$, which is what we were looking for.
I'll leave it to you to work out how the numerical value for the prefactor comes out to around $22.97$. You just need to plug in the various constants with the appropriate units.
If you are interested in a derivation of the expression for the rotatory strength, there is one given in Chapter 6 of Jeanne McHale's Molecular Spectroscopy.
