The first form of clapping originated in the 6th century BC, when Cleisthenes came to power in Ancient Greece as a democratic reformer and made behaviour like clapping a civic duty.
Because there was not enough time for every person to meet and greet their leader, clapping became a way in which they could do so, together as a whole.
Today, we clap not only for traditional reasons, but because clapping is a great equalizer. Studies have shown as opposed to vocalizations, which can betray a lot about the people who make them, subjects cannot get better than chance whether the clap they here is from a man or woman, nor can they guess the size of the individual, based solely on the sound of their clap.
Overall, clapping has become the standard form of admiration because it is by far the easiest, loudest and the most democratic. You cannot hear a wink or a thumbs up, not everyone can snap their fingers and clapping is less disruptive than stomping feet.