There's an ongoing discussing here between comments and answers about the definition of contradiction as asked by the OP. Since the question asked about the realm of logic, I'm going to provide an answer assuming that contradiction means the fact of statement A and not A to be held true at the same time.
OP asked why is it that humans cannot believe in contradictions, in the most literal sense of A and not A at the same time.
My take on it is that minds hold belief systems, complex networks of differently related facts with different certainty levels and relationships. This complexity of the networks is that gives rise to cognitive dissonance: facts that may have relationships to other facts which are contradictory indirectly.
The purpose of this network of beliefs is to be able to understand the world, predict it and interact with it.
Facts are as simple as logic proposes, little bits of knowledge of "X is true" and "Y is false". "X is true AND X is false" provides no information whatsoever, because nothing can be inferred from it, and endangers what related facts mean to it. A implies B, not A implies C, A and not A endangers the causality of A to B and not A to C.
This is why contradictions have this strange feeling into our mind: it endangers most of what we know. If statements are generic enough so that anything could be derived from them, contradictions on these statements are generic enough so that we may question everything we know. "This statement is false." is a great example.
(Notice I include paradoxes in contradictions, because the heart of a paradox is displaying one truth that contradicts itself.)