When Zakaria talks about the “rise of the rest” he is not talking about American decline in absolute terms. Instead, he is talking about how other countries that were once poor and weak are becoming richer and stronger both in absolute and in relative terms.
What Zakaria is saying in this book is that countries like India and China are rising. They are becoming wealthier and are even starting to become more internationally important. He notes that not all of the biggest things are in the US anymore. He mentions how the tallest building in the world at the time of his writing was in Taipei and the world’s biggest movie industry was in India. To Zakaria, these sorts of things mean that the rest of the world is becoming more competitive with the US. It does not mean that the US can no longer influence the rest of the world. However, it does mean that the US can no longer simply tell other countries what to do.