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What is the physical interpretation of chirality / chiral anomaly?

I'm dealing with this paper from C. Bär and A. Strohmeier about a rigorous derivation of the chiral anomaly. I'm not quite familiar with the physical context of chirality and its anomaly. What actually happens physically when an eigenvalues of the Dirac operator "flips" the sign? Are there any pictures like the Dirac sea explaining it intuitively? Are there any good references about that topic?