React.js and Dynamic Children - Why the Keys are Important

Recently I've been building a pretty dynamic interface based on google analytics data for one of our customers. There was a bug that I couldn't quite figure out because of my wrong understanding of how react works. In the end it turned out to be a very easy fix and very important lesson on understanding react.js. It is even pretty accurately described in react documentation however without showing what will go wrong and what kind of bugs you can expect if you don't adhere to it. In other words, the documentation explains the right way but without going much into details about why.