What's this do for us? First, it focuses our attention on scores that are positive, which, of course, result from actual matches (or in general from positively scored alignments). Long strings of mismatches and gaps are ignored. The idea then is to look for the highest score on the page and trace it back until you get to a zero. This will give you an alignment for a portion of each sequence - a local alignment. Then, if you wish, look for the second highest score that isn't part of the first trail of scores, and do the same. This will give you another significant local alignment.
One last word: what makes this better than a dot plot? Well, it's more rigorous, it allows us to use a more complicated scoring scheme, and it's something a computer can get its teeth into. That is, we can let the computer do all the work, ignoring these score arrays, and just get the best local alignments based on our dynamic programming algorithm. That's the way it works in practice, and the way it works when you hook up with a genome database.