Reich wrote Clapping Music for the same reason he wrote Music for Pieces of Wood, only in this case the instruments are even more portable: they come at the ends of the players' arms. Both performers clap out a rhythm. One of them then shifts it a beat, and goes on shifting it from time to time, till they are back in sync. If you try this out, you'll find it's not so easy. It's also not so easy to tell, just listening, that the two performers are clapping the same rhythm, more or less out of step, in each section.
About the composer - Steve Reich (born 1936)
All of Steve Reich's music has to do with repeating patterns that, as they go on repeating, gradually change. It's a fascinating way of making music, and worth trying out yourself-though you'll probably find it's not easy to come up with patterns as good as Reich's, or with repetitive processes as interesting. Often the ear is tricked by an alteration. You may think you're hearing something different, when all that's happened is that a pattern has jumped forward a beat, or a beat has been added.
Reich has spent most of his life in New York, and the pizzazz of the city is written into his music. But he's also been influenced by music from around the world, especially West African drumming, the gong orchestras of Indonesia and Jewish traditions of religious chant.