Mechanistic explanation for why the solubility vs. temp.

At equilibrium, always start with this equation:
delta(G) = delta(H) - T*delta(S)

Entropy is positive for the dissolution of a salt in water, therefore increasing the temperature T makes the overall delta(G) of the dissolution more negative.

Solubility is a simple equilibrium. NaCl <-> Na + Cl. Ignore the solid salt and the water as you always do in equilibrium expressions. Therefore K = [Na][Cl]. This means that the larger the equilibrium constant K, the more salt you'll dissolve.

delta(G) = -RTln(K), another important thermodynamic formula. As delta(G) gets more negative, K gets exponentially bigger: K = exp(-delta(G)/RT).

Therefore, increasing temperature will increase the equilibrium constant of dissolution, and therefore the solubility of NaCl.

Now, if you want to leave math out of it and just talk about mechanisms...when you drop salt into water, it exists as crystals of an ionic solid. In order to dissolve, water needs to "wrap around" ions and "pull them away" from the overall crystal. Water can do this because it's small and there's tons of it around (since, you know, it's the solvent), and once you pull an ion away from the NaCl crystal it's not going to go back.

However, to get that ion away in the first place, you need lots of "free" waters to continuously be hitting the crystal, and the waters that are solvating Na and Cl ions to be moving away quickly. Increasing the temperature of the solution makes the water move around faster, so this process happens faster. So that explains why you dissolve salt faster in hot water, but why can you dissolve more? Well, there's no good way to visualize this really, but it amounts to water molecules zipping around at high speeds being better able to accommodate big ions. Water normally has lots of hydrogen bonds to its neighbors (that are always being broken and reformed), but at higher temperatures those bonds don't last as long, so there's "more space" for an ion to fit in. That's...a really crude answer, which is why most of these physical processes are modeled with math.