From the beginning, he specifically chose to write for the general (non-technical) reader while avoiding conceptual simplification. This is significant, something few academics do. It exposes an important amount of detail of his subject, evolution, in a way that is approachable for all, yet permits analytical depth.
He further characterises his own writing as intellectual puzzle-solving, although I would say that much of the time he simply elaborates on a specific issue in evolution, worrying the finer detail. He also attempts to posit his treatises within a humanistic context - which is to say he often places the particular point in a wider human context of how scientific discoveries are wrought, and how social context interacts with those discoveries.
Yet most important is his treatment of finer concepts of natural history in non-jargon terms, such that any intelligent reader can see meaning without (indepth) training in the specific science.
I came to his writing by chance, one in a pile of library books once I was already delving into the subject. And his approach supercharged my enthusiasm.
Although he writes for the lay reader, if there is to be active pursuit there is no escaping the necessity to engage in background reading. Biology is the first place to start; included is genetics, paleontology, geology, and some anthropology, amongst other subjects.
It is also helpful to read in parallel. I am currently in the middle of his book Wonderful Life, a history and exposition of scientific discovery of the Cambrian era, close to the beginning of the emergence of modern animal forms. It is certainly a book of wonder, making an engaging narrative of a particularly dry subject like the classification of arthropods ( easily the largest grouping of animals, which includes insects, spiders, crustaceans, trilobites and others). The name arthropod, by the way, refers to jointed legs; the unifying feature is multiple body sections (some fused together in later evolution) with pairs of legs for each section.
However, where Gould sets forth four general groups of arthropods, the consensus-driven Wikipedia lists five - dividing uniramia into hexapods (insects) and myriapods (millipedes and centipedes), and placing the latter group closer to crustacean arthropods.
Part of the problem here is often that evolutionary biology is, well, an evolving subject, as much today as it ever has been, especially with the advent of genetic analysis.
Wonderful Life was written in 1989. A few scant years later, revision had already overtaken the work. A good example is the species Hallucigenia. In one of his essay subsequent to that book, Gould noted that that species had always been portrayed upside down - even as recently as his encompassing book on the matter.
So read a variety of works and authors on the subject - particularly the very recent. Yet having said that, I would be happy to own the full set of his collected works. The concepts are an intellectual challenge and a learning experience, and the books make a very enjoyable read.