Intuition behind the entropy definition
Pls. refer Pattern recognition and Machine Learning for details - section 1.6: Information Theory.



Left graph is Information(events) vs Probability. And the right graph is Expected Information/(Uncertainity) vs Probability. 

If we look carefully at the left graph; the variation at either extremes of the curve dies down, representing less change. Translated in current context, it means not much Information flows-in at extremes. 

Staying with the left curve, it's somewhere in the middle (45-degree, from vertex) where there's relatively more "meat" in the sense that more change happens/ translated, more information flows-in. 

Same thing is shown by the right curve: at two extremes, Change/ Expected Information is low at extremes, and highest at centre.

A crude Graph using Excel: