Nihilism is a philosophy of the proud. It isn't possible to renounce everything that has come before without a certain amount of confidence in one's own position. In Fathers and Sons, it quickly becomes apparent that the reason Bazarov is such a good proponent of nihilism is because he is incredibly vain. As much as the ideas of the younger generation, what will catch Pavel Petrovich's attention (Pavel is a proud man himself) is Bazarov's conceit. Indeed, it is – in large part – pride that makes the conflicts of the story escalate; they can't make themselves back down and seek reconciliation.
The reason that Pavel and Bazarov butt heads from the very beginning has less to do with their differing philosophies than with the fact that they are both immensely proud. Without vanity, they could quickly have come to a sense of mutual understanding.
When discussing his own insignificance, the key point Bazarov makes is that he is more aware of his insignificance than other people are of theirs. His newfound humility is, then, just another way for him to explain why he is superior to the people around him.