This was Yogi Bear's catchphrase. In his case, it meant that he was far more clever than the other bears in the park, which is why he wasn't satisfied eating nuts and berries - he craved the contents of picnic baskets!
The phrase has come to describe someone who outsmarts everyone else, or who somehow thinks their way out of a difficult situation. "Well done! You're smarter than the average bear!" By the way, when using this phrase, "average" is not pronounced in the traditional way; each syllable is emphasised equally and distinctly: "av-er-age."