In general, without an explicit explanation from an author of his or her purpose in writing a particular piece of literature, it is very difficult to understand with certainty the purpose or intention. Aside from the one obvious purpose--to entertain the reader--there exist many potential purposes, but, given the specific events in "A Rose for Emily," we can assume that Faulkner was interested in exploring several aspects of the human condition, specifically, repression and what evils repression can lead to; loneliness; the struggle between the individual and the larger society; and the struggle between the south of the Civil War and the rising modern south several decades after the Civil War.
In addition to our own assumptions about Faulkner's intentions, we are fortunate to have Faulkner's own words regarding his inspiration for writing the story. In a 1959 interview, Faulkner was asked what inspired him to write the story. Among other things, he said that he wanted to explore:
another sad and tragic manifestation of man's condition in which he dreams and hopes . . . another manifestation of man's injustice to man, of the poor tragic human being struggling with its own heart. . . .
Faulkner went on to note that the situation in the story was his invention but that repression in any form--in this case, Emily's repression by her father--was inevitably going to express itself "very likely in a tragic form." With Emily, of course, her father's selfishness in denying her the ability to have a husband, a home, and children created a woman who, in later life, would try to have at least some of those things but under completely twisted and tragic circumstances.
Essentially, then, Faulkner's intention was, in part, to explore the destructive effects of repression and to comment upon the power of a selfish father to destroy the future of his daughter, whose happiness he held in his hands, a betrayal of not only his position as Emily's father but also his duty as a human being to help another achieve a full life.