This can be answered by introductory chemistry class.

These are electrolytes solutions. The strong electrolyte has better conductivity.

What determines the strength of an electrolyte is the chemistry of the molecule, such as the polarity, inter-bond forces.

NaOH has three elements and two bonds. One bond is ionic and the other bond is covalent. When dissolved in water, only the ionic bond is broken, while the covalent bond between O and H remains intact, thus, OH ion is present.

On the other hand, NaCl has two elements and one bond. The polarity of this ionic bond is very strong. When dissolved in water, NaCl will dissociate to Na+ and Cl- ions.

Chemistry books have tables for strong and weak electrolytes, as well as tables for bond strengths. They will be helpful for you to understand why NaOH has higher solubility in water and why its solution is more conductive than NaCl.

Check out this free chemistry book online: https://openstax.org/details/books/chemistry-2e

Read chapters 6, 7, 8 and 11.

With all due respect I have to disagree with Mukhtar A Kareem Jaderson .

If you accept his explanation about the relative solubilities of NaCl and NaOH, you'll have a hard time explaining why LiOH is much less soluble than LiCl.

Dissolution is a process that can be defined by the free energy of the system before and after the process. It is not enough to look just at the salt; just as important is to look at the dissolved ions and their free energy of hydration (or more generally solvation, if the solvent is not water).

Molar free energies of formation, including for hydrated ions (in their "standard state") are available in tables, e.g. in the Handbook of Chemistry and Physics.

Some short-cuts are often possible. For instance, the melting point of a salt is pretty well correlated with its solubility - easy melting, easy dissolving. For the case of NaOH in particular, the capability to form extra hydrogen bonds also helps to increase solubility.

As for the high conductivity of NaOH solutions, this is partly due to a special mechanism similar to the Grotthuss mechanism that enhances proton conduction relative to other cations. See

https://water.lsbu.ac.uk/water/grotthuss.html#:~:text=Grotthuss%20hydroxide%20transfer&text=It%20has%20been%20proposed%20that,four%20electron%2Daccepting%20water%20molecules. A relevant excerpt follows.

Diffusion of hydroxyl ions

A similar process to that for hydrogen ions was initially proposed for hydroxide mobility:

Grotthuss hydroxide transfer

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However, it is now thought that hydroxide ions use an entirely different mechanism [371] for diffusion in an electric field. It has been proposed that the movement of the hydroxyl ion is accompanied by a hyper-coordinating (that is, a fourth hydrogen bond donor) water molecule. The hydrated hydroxide ion is coordinated to four electron-accepting water molecules. When an incoming electron-donating hydrogen bond forms (necessitating the breakage of one of the original hydrogen bonds), a fully tetrahedrally coordinated water molecule may be easily formed by the hydrogen ion transfer. The structure below left, HO−(··HOH)4, together with the more distant oriented water molecule below it, has been seen using neutron diffraction, with empirical structure refinement, of concentrated NaOH solutions [698]. The different mechanism involving additional hydrogen bond rearrangements plus reorientations, is the reason for the reduced mobility of the hydroxide ion compared with the oxonium ion. Interestingly, the transfer involves an anionic trimer (H5O3−), whereas hydrogen ion movement involved the cationic trimer (H7O3+) (note that neither of these trimers is stable by themselves). Density-functional theory-based molecular dynamics show that structural diffusion of hydroxide occurs mainly by a single hopping event [3225] due to the prevalence (≈ 50%) of the HO−(··HOH)4 species.

Hydroxide ion transport mechanism 📷

The H-O-D···−O-D 📷 H-O− ···D-O-D proton transfer in concentrated aqueous deuteroxide has been investigated using ultrafast infrared spectroscopy, which gave three picoseconds as the shortest time possible for the deuteron transfer kinetics. [2429]. Using two-dimensional infrared spectroscopy and ab iinitio molecular dynamics simulations, the dilute limit for the proton movement from one oxygen atom to the next takes 1.7 ps in pure water [4099].