The entire quote, I believe, is...
"Battle not with monsters, lest ye become a monster, and if you gaze into the abyss, the abyss gazes also into you." Nietzsche
The quote has many different meanings and applications, depending on how deep and subjective you want to go with it. I apply the quote to alcoholism as a reminder that by affording unnecessary attention to the thing that is killing me, I'm putting myself at an unnecessary risk for being killed.
It's a bit like a few other AA slogans, another one that comes to mind is... "Go to a barber shop long enough and eventually you will end up with a haircut."
A few AA cats told me the Nietzsche quote as wisdom to be remembered - after I asked about what I should do, considering my working in bars. It spoke volumes to me. As it applies to us booze hounds, it's just basically stating that you would be far more successful in avoiding harm by keeping the thing that wants to harm you far away from your proximity.