lithium chloride is considered to be an ionic compound, but it is soluble in alcohol, which suggests that it also possesses a small amount of covalent character.
This is a somewhat elaborate case, where the underlying point is that what solubilizes in alcohol is not the same as what solubilizes in water.
Water is a very small molecule with a very large dipole moment (intramolecular charge separation). The water molecule can easily accommodate charged species (ions), which it surrounds with its dipoles, forming stabilizing ion-dipole interactions. In addition water has a large dielectric constant, which results in screened interactions between ions that reduces their mutual attraction at longer distances. A third factor is that water has structure due to hydrogen bonding, which can change in character when a solute is incorporated, with associated changes in entropy and enthalpy. When the solvent becomes more structured, as happens when a highly charged ion is dissolved, the entropy of surrounding water is lowered, which discourages solubilization. However, as mentioned before, there are strong ion-water interactions which result in a large magnitude of the enthalpy of solvation (which is negative). Importantly, the devil is in the details of the relative magnitudes of the entropy and enthalpy changes.
Alcohols have neither as large a dipole moment nor dielectric constant (they are also not as structured by hydrogen bond networks, but that counts as a plus when it comes to solubilizing power). The net result is that ions are generally less soluble in alcohols. The strong mutual attraction of oppositely charged ions is not easily overcome by attractive interactions with the solvent. The result is a very low solubility of the ions.
But there is more (the catch). $\ce{LiCl}$ can solubilize as dimers composed of one $\ce{Li+}$ and one $\ce{Cl-}$ ion. The author interprets this as suggesting that there must be a covalent interaction that holds these atoms together (although why ionic interactions are not enough is not explained). The most important point here is that the energetics of dissolution of a covalent solid (for instance sugar) versus an ionic solid (table salt) are very different, because the strength of solute-solvent interactions are very different, as are solute-solute (or ion-ion) interactions. Ions interact much more strongly with the solvent and with each other, and in order to solubilize an ionic solid you need strong ion-solvent interactions that can compensate for the strong ion-ion interactions, if ion-ion interactions are indeed broken! In the case of $\ce{LiCl}$ breaking those interactions is not entirely necessary. In a covalent solid with small solutes, solute-solute interactions are weaker and solute-solvent interactions do not need to be as strong to compensate for disrupting the solid lattice.
Overall this is not entirely satisfying. If all this can seem confusing, I agree, there are a lot of details to unravel and some of the terminology sounds very similar (solvation versus dissolution, for instance). In any case, it is important to keep in mind that what happens in the solvent is only half of the picture. The other half is how the ions are organized and interact in the solid lattice. When dissolution of an ionic solid is discussed, it is often in terms of two contributions: the lattice enthalpy (which describes how strongly ions interact in the lattice, and the enthalpy of solvation (which describes how strongly the solvent interacts with the ions)*. If you ignore how strongly the ions attract in the lattice you will not fully understand why an ionic solid solubilizes.
*entropic factors can also be important as not all solutes dissolve exothermically (driven by an enthalpy change).
