Wikipedia's article on conjunctive normal form gives an idea of why a sentence is logically equivalent to its conjunctive normal form:
Every propositional formula can be converted into an equivalent formula that is in CNF. This transformation is based on rules about logical equivalences: the double negative law, De Morgan's laws, and the distributive law.
I think that what Russell and Norvig are getting at (the question is sort of vague, in my opinion) is that if you have a sentence like
$$ ((A \lor B) \to C) \land \lnot(D \land E) \tag{1}$$
you could then ask "in what cases would this sentence be false?" In this case, it's when
- One side of the conjunction is false, which means that
- either $(A \lor B) \to C$ is false, which means that
- $A$ or $B$ is true and $C$ is false, which means that
- either $A$ is true and $C$ is false
- or $B$ is true and $C$ is false
- or $\lnot(D \land E)$ is false, which means that
- $D$ is true and $E$ is true
From those you can read off a disjunctive normal form:
$$ (A \land \lnot C) \lor (B \land \lnot C) \lor (D \land E) \tag{2} $$
Those are all the cases where $(1)$ is false. If we negate $(2)$, it is like the "assertion that each possible world in which it would be false is not the case," and we get
$$ \lnot( (A \land \lnot C) \lor (B \land \lnot C) \lor (D \land E) ) \tag{3} $$
which by De Morgan's laws is
$$ \lnot(A \land \lnot C) \land \lnot(B \land \lnot C) \land \lnot(D \land E) \tag{4} $$
and then
$$ (\lnot A \lor C) \land (\lnot B \lor C) \land (\lnot D \lor \lnot E) \tag{6} $$
which is conjunctive normal form for $(1)$. So, by enumerating all the ways that a sentence could be false, turning that into a single sentence, negating it, and applying De Morgan's laws, we have a logically equivalent sentence in conjunctive normal form.