Why do solutions become saturated?
Take a slightly more general problem of miscibility, all that in thermodynamic equilibrium. In your question you mix up kinetics and equilibrium, and additionally in a wrong way. Equilibrium is the most important part. 

If you put two liquids together and shake them well, than in general, they do not mix. So you will observe a boundary between, say, diethyl ether (upper layer) and water (bottom layer).
If you separate them and analyze their composition, you will find, that the ether fraction contains little bit of water (1.5g / 100 mL) and that the water fraction contains little bit of ether (6g / 100mL). Knowing this, if you now take 100 mL of water and mix it with 5g of diethyl ether, you will observe, that no boundary is formed, and all ether has dissolved in water.

As another example, if you shake ethanol with water, they mix in any ratio.

That's observation, now the rationale. The particles of pure liquid interact with each other, in attractive way (otherwise you will have no liquid, but gas). Now you mix two pure liquids together (A + B) and look, if some of the particles of A can easily sneak into the B. If their behavior is reasonably similar, they can do it and interact with particles of B also in attractive way. The attraction should be comparable or greater, than within pure A. That is the enthalpy part.
But as we are dealing with free energy, there is also the entropy, which you have to take into account. On one hand, entropy favors mixing, as you add lot of accessible states. On the other hand, if the attraction of B to A is very strong, it starts to organize the molecules of A around itself, so they are not as free to move as they were in pure A, and limits the miscibility.

For the solubility of solids, it is technically similar, the A is solid and the interaction which hold sit together is the lattice energy.

As a side note, if you ask about solubility, miscibility and such, you should refrain from ideal approximations, as they do not predict such phenomena. Their calculation from pure substance properties is therefore terribly difficult, and physical chemistry tables are the only reasonable answer.