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It took me a number of readings before I figured it out, back in the day. I summarized my understanding in http://eli.thegreenplace.net/2010/01/02/top-down-operator-pr... -- hope it's helpful

I found http://journal.stuffwithstuff.com/2011/03/19/pratt-parsers-e... really useful as well.

The Pratt/TDOP scheme is a decentralized or OO way to organize the same idea. I coded them both, at https://github.com/darius/sketchbook/blob/master/parsing/pre... and https://github.com/darius/sketchbook/blob/master/parsing/pra...

(My Pratt is smaller than my precedence-climbing here because I ended up actually using the latter, so it got some extras added.)

Talking about the shunting yard algorithm, it's funny how often trains and railroads have inspired programming concepts. Parsing, concurrency and recently Scott Wlaschin describe monads through railways http://fsharpforfunandprofit.com/posts/recipe-part2/.

ps: while searching for Wlaschin talk I stumbled upon that https://www.google.com/search?q=monad+railway+equipment #rimshot