Having defined "complexity," Bonner presents his thesis: the complexity of the most complex living things has increased over time. The qualification is important, at often missing from similar theses propounded by, e.g., physicists and mathematicians. As Stephen Jay Gould puts it, the Age of Bacteria is not about to end any time soon. Bonner is fully aware of this --- indeed, as a specialist in the cellular slime molds, must have a most lively apprehension of it. Nonetheless, he does show that both plant and animal taxa which have evolved later have a greater number of cell types than their predecessors, and seeks an explanation acceptable to neo-Darwinism.
He finds it in the fact that complexity correlates very well with body-size, and for good reasons. To take what is perhaps the most fundamental point, if you are small enough, you can get oxygen to all your cells through diffusion, but above a certain size that won't work, and you need a specialized respiratory system. Similar things are true of digestion and other phases of metabolism, the control of movement, etc., etc., so if something is to get bigger, it will have to get more complex: and vice versa, increases in complexity (at least of certain kinds) allow an organism (or its descendants) to increase in size. Now there are solid and obvious advantages to being bigger --- you can exploit more resources (e.g., eat larger things), you are harder to kill, you have a more favorable surface/volume ratio: in short you are able to exploit new niches. If, later on, some of your descendants dwindle, to exploit a niche for small organisms, because of canalization they are generally stuck with your increased number of cell types, though Bonner points to a few instances which have become less complex: they are all, significantly, obligate parasites.
Bonner is logically led to consider the origins of multicellular life, and (what is separate) life-forms with specialized cell types. His speculations seem plausible, but are frankly beyond my competence to judge.
In summary, Bonner's argument goes as follows: Filling a previously empty niche does wonders for the reproductive success of an organism; variations which increase size make new niches available, and so are favorably selected. But, owing to constraints imposed by basic physics and chemistry, larger organisms must be more specialized internally, i.e. more complex, to be as efficient as smaller ones, or even just to survive, so the selection is especially favorable for larger and more complex organisms. Because of the way developmental processes work, this complexity will probably be retained even by later, smaller organisms in other niches. Voila: the evolution of complexity, by means of natural selection.