Faulkner doesn't present things chronologically.  Instead, he seems to organize things topically, and the breaks coincide with those topics.  The first topic is her death, the second her taxes, the third the smell, the fourth her father's death and Homer Barron, and lastly, the horrific discovery.  It is an interesting way to tell a story-topically-but it helps the reader piece things together just the same, and allows Faulkner to present all pertinent information in the order that best helps us to understand Emily's character by the end.

The breaks allow for this topical arrangment, and for the reader to be suspicious as they process things in-between each subject change.  Faulkner uses the subject changes and breaks to switch to the next relevant topic, and to give clues along the way to what happens in the end; that way, it makes a bit more sense to us.  He tells it more like a brain would think it; he mentions one thing, which reminds him of another thing because it directly applies, and by the time we get to the end, we have a full picture.  It also provides great entertainment as we go through and read it a second time, because we pick up on clues that weren't obvious the first time.

In "A Rose for Emily," why is the story divided into five sections?

Narrative structure is always important, particularly when we consider how it impacts our reading. The structure of a sentence, the choice of a word, the ordering of a paragraph can all influence what details we pay attention to when we are reading, as well as how we react to what we're reading.

The five sections that make up "A Rose for Emily" organize the nonlinear narrative in a way that allows a lot of very detailed information to be shared without overwhelming the reader. It's a good idea to read each of the five sections and determine what each is doing - what important part of Emily's story is it sharing? What foreshadowing is it employing? What bits of mystery is it unraveling?

It's also a fair point to consider that dividing the narrative, which is written from the plural first-person perspective of the townspeople, manages that collective voice as well. 

In "A Rose for Emily," what is the logic behind the story's division?

In his essay, "Another Flower for Faulkner's Bouquet: Theme and Structure in 'A Rose for Emily'," William V. Davis, writes,

Almost all of Faulkner's stories and novels can be better appreciated and more accurately understood and interpreted through a detailing of the interrelationships of time and structure. In Faulkner's world theme exists as the hyphen in the compound temporal-structure. Not the least of such cases is ‘‘A Rose for Emily.’’

With Emily Grierson, as for many other characters of Faulkner, the memory of the past is a powerful part of any present moment. Thus, the narrators of Faulkner's story return the reader to the past in order to illustrate and explain Miss Emily's character. For, it is in death that Miss Emily is presented to the reader as a "fallen monument" that in life "had been a tradition, a duty, and a care."  The tradition of the Old South is what has haunted Emily, a death that burdens her personality with its "imperious" demands upon her as a traditional figure of the community along with the death of her father which has formed her "inescapable" and "perverse" personality. 

This all-important interrelationship of time in the understanding of Emily Grierson's personality is what explicates her character, described by the narrators as "dear, inescapable, imperious, tranquil, and perverse."  Raised in the environ of the death of the South, living under the patariarchal influence of her dead father, Emily, having lived with death all her life, becomes enamored of death and, therefore, "perverse," she weds death as she lies with the cadaver of Homer Barron.