I was lucky enough to work with Prof. Gelman as his research assistant while I was in school - I can't even being to tell you how prolific and brilliant that man is. His name may not be known very much outside academic circles, but I'd go as far as to say that he's the most important Bayesian statistician since Thomas Bayes.
He used to be a contributor to FiveThirtyEight, back before the Times picked it up. I used to explain FiveThirtyEight as 'one of the six blogs Andrew Gelman writes for'. Now, I explain Andrew Gelman as 'a former contributor to Nate Silver's blog'. How times have changed!
Gelman's approach to statistics is more wholly Bayesian than most people with a moderate level of statistical training are likely familiar with. It was from Gelman that I learned why I never need to perform an F-test[0]; at the same time, it was from Gelman that I learned some of the potential pitfalls of pure Bayesian reasoning[1] (and how to address them).
When people ask me where to get started with statistics, both of the books I recommend are Gelman's: Teaching Statistics: A Bag of Tricks and Data Analysis Using Regression and Multilevel/Hierarchical Models.
Both have tremendously off-putting titles, but they're actually incredibly accessible. Gelman is great at many things, but picking sexy titles is not one.
If you're interested in understanding the concepts behind this paper, I'd start there.
[0] http://andrewgelman.com/2009/05/18/noooooooooooooo/
[1] The linked paper provides a good analysis