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.