If you thought Calvino was just being careless with all the fragmentation in plot and narration, you're just flat-out wrong. This guy—just like his trickster character, Ermes Marana—knew exactly what he was doing. Fragmentation is one of the faves of postmodernist authors, and Calvino falls squarely into this group.
Postmodern lit also seeks to explore how literature goes about representing stuff that just can't be represented—which might actually be everything in the real world. Calvino knows that readers expect a very clear and straightforward plot, but insists on giving us almost the opposite: a plot that constantly starts, then stops, then goes off in some other direction.
In this sense, the book is also an example of a genre known as metafiction, which refers to books that comment directly on the fact that they are books, or perhaps stories that contain other stories within them (Calvino's book does both).
Calvino is suggesting that we rethink the ways that stories are "supposed" to be told. After all, if we keep getting exactly what we expect when we read books, how are we going to keep an open mind to new possibilities? What this novel attacks above all else is complacency in reading—the desire to get exactly what you expect when you sit down to read.
It is only by blowing tradition apart (enter postmodernism) that we can find new sources of energy in storytelling, and Calvino wants to help you—as gently as he can—to do the uncomfortable work of changing the way you think about reading.