The actual word for Emily is not that she went crazy, but that she "snapped". And for this, she had many reasons.
First, she lived her entire life under the spell of her overwhelming father, who controlled nearly every aspect of her life, and limited her immensely.
Second, she never learned to live on her own, nor to be self-sufficient without feeling insecure. She had enormous gaps in her psychological development which made her unable to establish social bonds, and even more unable to love and be loved.
Being unable to connect, whatever connections do come along become a huge deal that is nothing like the feelings normal people get (normal is a term used loosely, granted). Emily saw in Homer the first and only chance to connect, which is a human need, and the moment he was about to leave her, life as she knew it would have fallen under her feet. She had to do something, and she thought of the ultimate blockage: Death- This is when she snapped and got the arsenic to kill her only link outside her inner circle of dysfunctional family members before it left her first.
In my opinion, Miss Emily goes crazy because she lacks affection -- no one loves her. The one who is to blame for this is her father.
During her youth, Miss Emily's father is said to have pretty much chased away any man who wanted to have anything to do with Emily. This meant that Emily became relatively middle aged and was still single. So when she met Homer, he was her last chance.
When he didn't want to marry her, she went over the edge. Overall, then, I blame her father for making it hard for her to find love.
To add to the responses above, I do not think that "crazy" is quite the right term for Emily. She does engage in a despicable act with Homer Barron's body after she murders him; however, I think she was fully aware that her actions were wrong but felt compelled to do them anyway. Regardless, she is mentally unstable, and although her father is certainly partially at fault because he raised her with such rigid rules, Emily is ultimately the one responsible for her own actions. Emily's father is long dead by the time she meets Homer Barron, and although her father's principles may have lingered after his death, Emily has the capacity to live under her own rules. She, however, chooses not to help herself and is therefore to blame for her own actions.
I wouldn't say that Miss Emily Grierson was crazy, but she is certainly the very eccentric protagonist of William Faulkner's short story, "A Rose for Emily." Emily has had little luck with men over the years. She has been dominated by an overly strict father and then ditched by her Yankee beau, Homer Barron. Her decision to buy the rat poison is a calculated one, and her choice of living a life of seclusion is not necessarily one of madness. Once Homer has left her and her youth has advanced to middle age, she apparently finds little reason to leave her house. Since she never socialized with many of the townspeople in the past, she has few friends and makes no new ones. She chooses to just fade away inside her aging home rather than face a new life in the unfamiliar and changing world outside.