On slide 12, you can see that SAT(φ) holds iff not VALID(¬φ) holds. Hence, to use the sequent calculus (tool) to generate a satisfying assignment, you try to prove that ¬φ is valid. If that proof fails with a counterexample to ¬φ, you have a satisfying assignment for φ. See also slice 108 of the propositional logic chapter that discusses that in detail.