Standardized Selection Ratio (SSR)
I can see why you might be confused! The authors chose bad notation, since the way they wrote it $w_i$ clearly depends not just on the choice of anenomefish species $i$ but also on the choice of anenome species $j$. And it is not clear how exactly they are performing their sum in the definition of SSR: sum over all possible anenome-anenomefish species pairs or only over all anenomefish species? The former makes sense but is extremely unclear, and the latter doesn't make sense because there's no way to interpret that as a probability that an anenomefish will choose a particular anenome because the sum of the SSR for all the different anenome species would not necessarily be 1, so the probability of an anenomefish species choosing some species of anenome would not be 1! 

NOTE: The reference to Manly 1993 was incorrect (you cited a 1995 paper); they actually cite a book: Manly B, McDonald L, Thomas D (1993) Resource selection by animals. Statistical design and analysis for field studies. Chapman and Hall, London. This would probably be the best place to look to get an exact answer to your question. I think that would be the best place to look first.