On page 33 of these notes by David Skinner, it is claimed that
[starting from a connected graph and removing the bridges] tells us how to compute $\Gamma(\Phi)$ perturbatively from the original action: $\Gamma(\Phi)$ consists of all possible 1PI Feynman graphs that may be constructed using the propagators and vertices in $S(\phi)$.
However, I cannot decipher exactly what this means. How does one go about computing $\Gamma(\Phi)$ using Feynman diagrams as described? By writing out the 1PI Feynman diagrams, should I not just get a number, rather than the effective action with an explicit $\Phi$ dependence?
EDIT: I have read Proof that the effective/proper action is the generating functional of one-particle-irreducible (1PI) correlation functions, but I do not understand how this allows us to directly calculate $\Gamma$?