The direction from a satisfying assignment is by induction along the paths: an edge preserves truth, so a true literal reaches only true literals, and a variable cannot be true together with its negation. The other direction builds an assignment. The graph is skew-symmetric — an edge $a \to b$ gives an edge $\lnot b \to \lnot a$ — and so a set of literals closed under successors that never contains a literal together with its negation can be extended by any variable not yet decided: one of its two literals does not reach its own negation, and that literal together with everything it reaches joins the set consistently. When every variable is decided the set is a satisfying assignment.

The empty clause is stated separately because it has no literal and so no edge; the criterion about paths says nothing about it.