I believe that the authors, of the reference you provided, explain the reason behind introducing these so-called quasi-Fermi levels at the end of section 4.3.3. For simplicity, let me just repeat it here; perhaps explaining it in different words, with a little more elaboration, would help. I’m sure you're aware of the fact that in $n$- ($p$-) doped semiconductors the Fermi level is closer to the conduction (valence) band as opposed to being close to the middle of the band gap. To be more mathematically precise, the Fermi level of $n$- ($E_{F,e}$) and $p$-doped ($E_{F,h}$) semiconductors is given by $$E_{F,e}=E_{F,i}+k_{B}T\ln\left(\frac{n}{n_{i}}\right)$$ and $$E_{F,h}=E_{F,i}-k_{B}T\ln\left(\frac{p}{n_{i}}\right)$$ respectively. The quantities $E_{F,i}$, $k_B$, $T$, and $n_i$ are the intrinsic Fermi level, Boltzmann constant, temperature, and intrinsic carrier concentration respectively. The quantities $n\approx N_{D}-N_{A}$ and $p\approx N_{A}-N_{D}$ where $N_A$ and $N_D$ are the acceptor and donor concentrations.
Now, when you have a steady light source irradiating your semiconductor sample, you create a large number (compared to thermal excitations) of Electron-Hole Pairs (EHPs). You could think of it as doping of the sample with equal number of electrons and holes; let us call this “photo-doping.” In this photo-doping picture it is much easier to think of two separate quasi-Fermi levels for electrons and holes. You can notice that the above two equations are nothing but a rearrangement of equation (4-15) from the reference you provided. The reason they are called “quasi”-Fermi levels is because the Fermi level is only defined in equilibrium. But in this particular case, steady-state could just be treated as a different type of equilibrium.
To answer your question as to why we don’t have two quasi-Fermi levels in equilibrium I would say: in principle you could define quasi-Fermi levels even when we have EHPs due to thermal excitations. However, the typical concentrations of thermally excited EHPs are so small that it can be incorporated into the broadening of the Fermi-Dirac distribution function. In the mathematical sense, you could also incorporate the case of photo-excitation (or photo-doping) into the broadening of the Fermi-Dirac distribution as well. But then you run into the physically counterintuitive problem of defining what temperature means. First of all, a sample irradiated with light is not under thermal equilibrium. Secondly, since temperature, in the conventional sense, is associated with thermal energy of the system, it makes sense to only incorporate thermally generated EHPs into the broadening of the Fermi-Dirac distribution.
In your particular example, we have an $n$-doped system. Consequently, the concentrations $n_0 = 10^{14}$ cm$^{-3}$ and $p_0 = 2.25 \times 10^{6}$ cm$^{-3}$ exist under thermal equilibrium. When (say) you shine light on it, we have $n \approx p = 2 \times 10^{13}$ cm$^{-3}$ due to photo-excitations. In this particular example, the quasi-Fermi level of electrons is not that far away from the Fermi level in equilibrium. The quasi-Fermi level of the holes, however, is significantly away from the Fermi level at equilibrium. This is a direct consequence of the fact that the concentration of minority carriers jumps by a large amount. Now, if we were to define separate Fermi-Dirac distribution functions for electrons and holes, under this steady-state condition, and we computed the respective Fermi-Dirac integrals using the density of states of the conduction and valence bands, then we would obtain the correct electrons and hole concentrations (i.e. $2 \times 10^{13}$ cm$^{-3}$). This is another way of justifying two separate quasi-Fermi levels.
It makes sense why the separation between the quasi-Fermi levels would be proportional to the rate of EHP generation due to photo-excitation. As a result, the separation between the quasi-Fermi levels, just as the authors say, is a measure of how much the system is out of equilibrium. So, in summary, I believe that the best way to get an intuition for quasi-Fermi levels is by considering the photo-doping picture. Although the authors don’t mention this explicitly, I have a hunch that the concept of quasi-Fermi levels was inspired from conventional doping (using donor or acceptor atoms).
Best Answer
A diode consists of two materials known as p-type and n-type semiconductors, connected in series which allows current to flow through them differently. In the n-type semiconductor, electrons travel with enough energy such that they're not attached to an atom and are said to be in the conduction energy band. For the p-type semiconductor, electrons "hop" from atom to atom, but lacking the energy to free them, are said to be in the valency energy band.
At the interface between the n-type and p-type materials, a travelling electron has to move either from the n-type to the p-type in one direction, the p-type to the n-type in the other, to continue moving. Is there a difference between the two directions?
Well, an electron moving from the n-type to the p-type material can occur spontaneously because the free electron's energy is released as radiation and it can move to a lower energy state, attached to an atom in the p-type semiconductor. But to move from the p-type to the n-type it has to gain energy from somewhere, and this isn't spontaneous because there is no guarantee of some other process providing this energy.
Think of a ball at the top of a hill: It can move from the top to the bottom while spontaneously releasing energy, but some other process must provide the energy to take it from the bottom to the top. And this analogy provides a basic explanation for why a diode conducts in one direction, but not in the other.