This book is about the scientific method as much as it is about the Royal Society. It starts with the stories of four men who Gribbin considers were the first practitioners of what we call the scientific method - hypothesis, experiment/observation, revision of hypothesis if necessary. These are William Gilbert, Galileo Galilei, Francis Bacon and William Harvey.
From there, it tells the story of the Royal Society from its beginnings in 1660 to the mid 18th century, but mostly concentrating on the 17th century. On the way we get brief biographies of many of the men involved, including Boyle, Hooke, Newton and Halley. This is a good starting point if you don't know much about these famous figures in the history of science, as it helps to put them all in relation to each other.
It is very readable - recommended.