What does "damn near" mean?
And she stared up at me with a look of such profound sadness it damn near broke me in half and said, quietly, “Thank you for coming.” I somehow replied, “It’s because of your son that I am standing here.”

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Near in this case is used as a adverb meaning ‘ Just about; almost; nearly’

We often use damn to emphasize a statement, 
Ie., I am damn sure this is correct. 
     Damn sure means very very sure,
     The story is damn funny
   Damn funny means very very funny.

So taken together damn near means very nearly, very almost. very near the point of something that will change something.

Hope this can help you . Thanks

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yes, J. Rios is right: it means "almost," in this case.

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almost