Hello Rex.  I'm new here too.  This is my first comment.

I cheated and did a Google search for that quote.  
Here's the link to what Yahoo Answers says: http://answers.yahoo.com/question/index?qid=20080803092259AAJsGHP

But my first reaction to that statement is that once you concentrate on the abyss, the abyss that is the mystery of why anything is here, a strange feeling can engulf you.  Suddenly everything does becomes strange, and you say to yourself, "My God, why am I here at all??  Why is anything here at all??"   This is the profound feeling, "Why is there something rather than nothing?", the question that the philosopher Martin Heidegger takes as the central problem of philosophy I think.

The above may not have been what Nietzsche meant by this particular statement, but it was the first thing that came to my mind. 

Best regards,
Marden

Hi Rex, Marden,
Interesting question and response. I have a different interpretation which is influenced by the book of Susan Sontag "Illness as a Metaphor." She takes the most frightening subject and analyzes it in depth. While reading it I was struck by my own fear while reading it and by her courage. Especially when you know that she was ill herself while writing it, her courage is doubly impressive and I was full of admiration at her ability to look the abyss in the eye. So my interpretation of the phrase you give Rex, would be that once you get your courage up to stare at the abyss, it stares back at you in the sense that looking at it with courage demystifies the abyss itself and makes it equal to you. From a black hole it is something that has a stare, something you can engage with without it overcoming you.