"The capacity to doubt herself, to abandon - albeit in the face of strong resistance - her self-assurance and self-satisfaction, lies at the heart of Europe's development as a spiritual force. She made the effort to break out of the closed confines of ethnocentricity, and her ability to do so gave definition to the unique value of her culture. Ultimately we may say that Europe's cultural identity is reinforced by her refusal to accept any kind of closed, finite definition, and thus only affirm her identity in uncertainity and anxiety."
"Looking for the Barbarians: The Illusion of Cultural Universalism" (1986)
What does it mean to be self-critical? In the Catholic tradition, we have what's called an "examination of conscience": a means of reflecting on our lives to see where we've gone wrong or to try and reconcile hard decisions with our values. It presupposes a couple things: that there is a God, that we possess a human nature, that there are ways to offend God and our own human nature, and that this rupture betrays that nature. The result of this is guilt, bad consequences, and redefinition of our own nature. To offend conscience is to recreate ourselves instead of allowing a Creator to continue to sustain His creation and to develop you in an ongoing act of creating.
This all presupposes the ability to think and reflect. Hannah Arendt defined "imagination" as the process in which we are able to dialogue with ourselves. Under that is some concept of a higher good or criteria that our actions are subservient to. To make ourselves and our own actions that highest criteria of right thinking or right action, in a sense, obliterates the dialogue because, at some level, one side of the interior dialogue is suppressed and the other is allowed to dominate. That suppressed side of the dialogue is most likely the one that doubts and criticizes actions that are (or allowed to be seen as) ambiguous.
But that act of dialogue with oneself also presupposes a few things too: a sense of self, the concept of an "other" (even if that "other" is a projection within oneself), the superiority of one view of another, and a final, binding judgment. Anxiety may come from a suspension of judgment or even the rejection of that judgment but, at some level, it presumes the validity of the idea of judgment. Even when someone is considered too critical of themselves, there is a "self" that is being judged and the resulting anxiety testifies to both the presence of an interior dialogue and an imbalance that seeks equilibrium.
When that interior dialogue is happening - so long as it actually is a dialogue - the acceptance and use of judgment, discernment, and even "otherness" presupposes something higher than the point of views, an ultimate "criterion", however elusive.
To avoid presuming a predictable conclusion for too long, this is one way to see how that the idea of an "ultimate" criterion that aids in discernment and judgment is a helpful way of understanding a tacit acceptance of the idea of "truth" that we all seem to have. Some people feel that this idea of "truth" is a useful way that our minds manage the things we think about, a sort-of silent party that allows for the possibility of a final judgment but for many others its a witness to the way God organizes our conscience and to conceptualize the reality of transcendence, that which exists beyond ourselves and our judgment.
Kolakowski's point in the quote above sees that the heritage of self-criticism is itself a sign of civilization and (to run the risk of a kind of triumphalism) is also a sign of a higher sort of civilization. As such, it's also a higher sort of humanity, a sign that something is going right and a principal for growth.
Though it's natural to avoid guilt and coming up short in examining our lives and failures, it's also "supernatural" to embrace self-examination. The tension and the anxiety is a sign that, at some level, something is going right. Better to go through it than walk past it.