Around Meiji era (1868-1912), the form of 二礼二拍手一礼 “two bows, two claps, and one bow” started to become the official form of worship when visiting a shrine. It became the official form through out the Chain of Islands of Japan.
When you come to the front of a shrine, you first bow twice then clap your hands twice, and bow again at the end. As for the priests, the numbers of clapping hands may differ due to the tradition of the area. For example, the Izumo chain shrines where God Kunitsu-kami (国津神) is worshipped, the formal way is to clap four times. The formal way in Ise Jingu or The Grand Shrine of Ise is to clap four times x 2, which makes it eight times. It is called yahirade(八開手).
I am sure that Izumo Shrine has their own story of how it became four claps, but in my vision I see that it is worshipping the Four Gods of Purification, Haraedo-Yonshin (祓戸四神) lead by Susanoo. As for the eight claps of the Ise method, if you count four claps as one, it becomes two that is the original idea of the official form of today. In Shinto, the number eight has always been a sign of good omen since the ancient times.
People express their joy and excitement by clapping hands. It is an act of instinct seen all over the world. Babies clap their hands when they are having fun. It was the same in the ancient times. The ancient people clapped their hands to express their gratitude and sense of awe by the presence of god. They were actually able to see the gods. At an Iwakura (磐座)) - a landing place for the gods in energy form, the ancient people witnessed the energy form twisting and swirling up like it was dancing. The people naturally start clapping as they watched this magnificent thing happen. It seems that how many times you clap wasn’t really so important.
In my personal opinion, the reason why the offical form of worship settled to “two bows, two claps, and one bow” after much meandering, is because the “two bows” represents Izanagi and Izanami who are a husband and wife couple gods, and the official two gods who are the originator of the later gods. The “two claps” signifies to show respect to the two gods. The last “one bow” signifies that your innate god is one with the originator. It is a matrix to be in oneness.
Thank you for letting us live