The (¬ϕ→¬ψ)→(ψ→ϕ) [CCNpNqCqp in Polish notation] axiom can get interpreted as meaning that if a negation of a variable is equivalent to the negation of another variable, then the variables will be equivalent.
(¬ϕ→¬ψ) says that if the negation of the first proposition is true, then the negation of the next proposition is also true. Given the rule of substitution as presupposed, that implies the negation of the next proposition implies the negation of the first proposition (¬ψ→¬ϕ), making the negation of the first proposition equivalent to the negation of the next proposition.
Likewise, if (ψ→ϕ), then (ϕ→ψ) follows.
So, the fourth axiom in some sense gives a sort of cancellation law for negations. And in some sense, negation, semantically speaking, just tells us to cancel out one truth value and replace that truth value by a specific other truth value. If we only have two truth values, that means we replace one truth value with another. Thus, intuitively speaking the fourth axiom manages to tell us everything about negations by giving us a cancellation law, and negations just cancel something.
Laws like Pierce's law CCCpqpp come as a bit more difficult to explain intuitively, though here goes. First let's observe that the implicational propositional calculus can get axiomatized using the axiom set
Thus, we only need to explain how the axiom CCNpNqCqp can derive Pierce's law. Well, since we already have all tautologies with negations, we have the law of Clavius: CCNppp. Rewriting that with the definition Np := Cp0 we obtain CCCp0pp. Thus, since we have the law of Clavius we have a special case of Peirce's law, since we can replace "q" in CCCpqpp to obtain CCCp0pp. We also have CNpCpq. Thus, if we assume CCpqp and Np, we'll end up with with the ability to derive p, and thus CNpp without Np as an assumption, and thus p follows by the law of Clavius. So, in some sense we can derive Pierce's law and thus all other tautologies, because CCNpNqCqp enables to find "the key" to prove all the other tautologies.
Note also that in addition to CCNpNqCqp, the much stronger EENpNqEqp holds.