I use that sheet as follows. If I'm logging into Amazon I'll find the intersection of column M and row A (the second and third letters of Amazon) and then read off diagonally 16 characters. That would be my Amazon password (in this case, TZ'k}T'p39m-Y>4d); when I hit the edge of the paper I just follow the edge).
The security of this system rests on the randomness of the generated characters and the piece of paper.
PS Yes, it's a total pain to use long, random, different passwords.
PPS If it's not obvious to people you can add a second factor to this (something only you know) in the form of the algorithm for picking the password from the sheet. For example, instead of using the second and third characters from the site name you could pick any combination. And you could change the letters as well (e.g. for Amazon you could use the last two letters moved on one place in the alphabet; you'd have PO as the key). Also you don't have to read diagonally but could use any scheme that works for you (e.g. a spiral pattern, read vertically, read characters at offsets from the start based on the Fibonacci sequence, etc.).
