Probabilistic Intuition behind connected correlations and 1PI vertex function

In probability theory $W=\log Z$ is called the generating function for cumulants. Its Legendre-Fenchel transform $\Gamma$ is the rate function appearing in theorems on large deviations such as Cramer's Theorem. I think if you want more information about the probabilistic meaning of $\Gamma$, then you should lookup the literature on the theory of large deviations. A classic book on the subject is "Large Deviations Techniques and Applications" by Dembo and Zeitouni. A nice introduction with emphasis on statistical mechanics is "A Course on Large Deviations with an Introduction to Gibbs Measures" by Rassoul-Agha and Seppäläinen.

Replace in your formula $S,W,\Gamma$ by the same letters divided by the Boltzmann constant $k$ (the probabilistic analogue of Planck's constant $\hbar$). Then, in the limit of $k\to 0$, the effective action becomes the original action, $\Gamma=S+O( k)$. Thus the effective action is the low temperature (in statistical mechanics ) resp. semiclassical (in quantum field theory) analogue of the (zero temperature, resp. classical) action, and can be determined by an expansion in terms of $k$. As it contains all information about the probability distribution, it is a very useful quantity.