Heh, this is interesting. What do you think about this phenomena? I lurked a bit and found some explanations, but feel free to give your own.
1) "Clapping originated in 1473 during the original outbreak of Cholera. At its origin clapping was not an act of applause, rather it was used to tell others whether you were infected. Over time clapping evolved into a way of showing appreciation and happiness. In 1876 clapping continued its evolution appearing in music and as a form of percussion."
2) "No one knows the origin of clapping to express appreciation/enjoyment, but it's been around since at least the Middle Ages (Chaucer wrote about it). If you want to go back even further, the Bible (in English, at least) says "Oh, clap your hands, all you peoples! Shout to God with the voice of triumph." This could be construed as applauding a job well done. Clapping also seems to be an instinctive gesture; babies often do it when they're happy.
However, clapping to express appreciation is not exactly universal. Some cultures favour foot-stomping, whistling, or simply yelling. And even in clapping cultures, clapping doesn't always mean "bravo!" Sometimes it's done to keep a beat, sometimes it's done to frighten away animals. So clapping after good performances is most likely just a custom that we learn from our parents and our society, and pass on to our children.
One thing in clapping's favour is that it's probably the loudest possible non-verbal noise you can make with your body. Your message of approval is more likely to reach the performers than something quieter, like finger-snapping or armpit farts. And because it's not verbal, it doesn't matter which words you say or even which language you speak. Everyone can clap together."
3) "The custom of applauding may be as old and as widespread as humanity, and the variety of its forms is limited only by the capacity for devising means of making a noise (i.e., stomping of feet or rapping of fists or hands on a table). Within each culture, however, it is usually subject to conventions.
The ancient Romans had a set ritual of applause for public performances, expressing degrees of approval: snapping the finger and thumb, clapping with the flat or hollow palm, waving the flap of the toga, for which the emperor Aurelian substituted handkerchiefs (orarium) that he had distributed to the Roman people. In Roman theatre, at the close of the play, the chief actor called out "Valete et plaudite!", and the audience, guided by an unofficial choregus, chanted their applause antiphonally. This was often organized and paid for.
Similarly, a claque (French for "clapping") was an organized body of professional applauders in French theatres and opera houses who were paid by the performer(s) to create the illusion of an increased level of approval by the audience.
With the proliferation of Christianity, customs of the theatre were adopted by the churches. Eusebius says that Paul of Samosata encouraged the congregation to applaud his preaching by waving linen cloths (οθοναις), and in the 4th and 5th centuries applause of the rhetoric of popular preachers had become an established custom. Applause in church eventually fell out of fashion, however, and, partly by the influence of the quasi-religious atmosphere of the Wagner performances at Bayreuth, the reverential spirit that inspired this soon extended back to the theatre and the concert hall."