How does humanity win over sin in the story "The Tell-Tale Heart"?

I like the word humanity in your question--in order to
answer this question you have to define what you mean by humanity.  We tend to think of
humanity as our kinder, gentler nature, but in the case of this story, you have to think
a little more.  It is hard to consider that the man who killed the old man has any
humanity.  Yes, he is human, but he cold bloodedly kills the old man for no good
reason!  And yet it is his humanity that triumphs in the end
because in this story, humanity means conscience.  Most human beings have a conscience
that is their moral compass.  For a while, the man seems to not have a conscience, but
in the end, the guilt that comes from his conscious shows his humanity.  After he kills
the man and dismemebers the body he imagines he can hear the beating heart of the
victim--those heart beats are his guilty conscience condemning his actions and his guilt
overwhelms him into a confession of his sins--thus the murder is revealed and his
humanity wins over his sin.