Thermodynamics behind organic chemical reactions?
Whether or not the product is "more stable" would really depend on the sense in which you consider its reactivity. The $\mathrm{p}K_\mathrm{a}$ of a $\beta$-keto ester is about 11, so in that sense, it is much more reactive than either of the reactants. Once the $\beta$-keto ester is formed it is quickly deprotonated. It remains predominantly in this form because the anion is so stable, and the reaction is incapable of reversing. Thermodynamics does not always determine the result of an organic reaction. Many times, kinetics is the deciding factor.

The Claisen condensation is catalyzed by the base, as it is recovered in the end, and though I am sure it occurs spontaneously without it, I doubt it happens very often. I don't think ethyl acetate would be as popular a solvent as it is if it quickly and spontaneously converted to ethyl acetoacetate.