If you understand the Gibbs energy concept in general, then it is no different for the specific case of solubility. The spontaneity (ΔG) of any reaction/process depends on both the overall enthalpy change and the entropy change. In the case of dissolution of a crystalline solid, the ΔG will depend on (1) whether net heat is released or absorbed during breaking of the crystalline lattice and formation of new intermolecular interactions between dissolved components and the solvent; and (2) whether a net entropy is positive or negative during breaking of the ordered crystalline lattice and formation of hydration spheres around the dissolve solute components. All of these factors are not always easy to predict, and which ever one wins out will determine whether dissolution happens spontaneously at a given temperature.
If you do not understand the Gibbs energy concept at all, then that would be a good place to start, ignoring for a moment the specific case of dissolution.
It's really important to also understand that even if ΔG is negative, this only means that the process will eventually proceed in the forward direction toward an equilibrium point that favors formation of products. This says nothing about how long it takes to get there. The world is full of thermodynamically favorable processes that never happen over reasonable timescales because they aren't kinetically favorable.