In Hard Times, what is the significance of the book structure?
This is a great question! Well done for noticing the curiously entitled headings given to each section of this great novel. I guess one way to approach this question is to consider a wider theme within the novel which is that of clocks and time. Throughout this text, mechanical time and natural time is juxtaposed and contrasted. In Coketown and in the household of the Gradgrinds, time is severely mechanised - it is monotonous, structured and inexorable. Consider such quotes as "Time went on in Coketown like its own machine." Think too of how this concept is symbolised by the "deadly statistical clock" in Gradgrind's study.
However, opposed to this ruthless view of time is the structure of the novel that you have identified that divided the overall action by natural time. Thus the three book titles, "Sowing," "Reaping," and "Garnering," refer to agricultural labour and to the natural processes of planting and harvesting in accordance with the natural passing of time in the seasons. Likewise the narrator comments on the change of seasons even in Coketown's miasma of smoke and red-brick buildings. According to the narrator, these changes of season amount to "the only stand that ever was made against its direful uniformity."
Dickens seems to create this contrast throughout this novel with the purpose of emphasising how industrialisation has mechanised the human experience as well as heightening the opposition between agricultural labour and labour in a town such as Coketown, which, as we can see through characters like Stephen, only destroys and grinds down humanity.