The "joy that kills" is Louise Mallard's ruined dream of experiencing a free life the moment she discovers that Brently is alive. When Louise Mallard learns that her domineering husband, Brently Mallard, has tragically died in a railroad accident, she is initially overcome with grief and goes upstairs to compose herself. Once Louise is alone in her room, she begins to think about her future as a widow and realizes that she will be completely free and independent. Louise recognizes that she will have the rest of her life to do as she pleases and will no longer live under her husband's forceful hand. As an independent widow, Louise can experience life without any restrictions and is free from her oppressive marital obligations.

Louise's feelings towards her husband's death change when she sees the silver lining of the tragedy. Chopin writes that Louise is "drinking in a very elixir of life through that open window" and that she "breathed a quick prayer that life might be long" before she leaves the room. Once Louise walks down the steps, she discovers that her husband is alive and suddenly dies of a heart attack. Although the doctors assert that Louise died of a "joy that kills," the reader recognizes that Louise's heart broke the moment she realized that her hopeful, independent future was destroyed. Therefore, the "joy that kills" is Louise Mallard's ruined dream of being free and living for herself.

In Kate Chopin's celebrated short story "The Story of an Hour," Louise Mallard receives the tragic news of her husband's sudden death in a railroad accident and she immediately begins to weep with "wild abandonment." When Louise retires to her upstairs room, she recognizes the beautiful spring weather and begins to contemplate her future as a widow. As Louise thinks about her future, she experiences an epiphany and is overwhelmed with the possibility of hope, freedom, and joy. Louise recognizes that she will no longer experience an oppressed, stifled life, where she is forced to bend to the will of her husband and live up to society's expectations. While Louise sits in her chair, she cannot contain her excitement and repeatedly says, "Free! Body and soul free!"

As Louise descends the stairs with a feeling of renewal and hope, her husband enters the house and she dies from a heart attack. The doctors attribute Louise's heart attack to a "joy that kills." They misinterpret Louise's feelings and believe that her joy stemmed from Brently being alive. Ironically, Louise's joy stems from Brently's death, which leads to her brief moment of liberation and the heart-warming realization that she will have an independent, autonomous future. Brently's arrival completely shatters Louise's future and the terror of losing her newfound freedom caused her heart attack. Therefore, the joy that kills is actually the exciting feeling of being an independent widow, which is suddenly taken from Louise when she discovers that her husband is still alive.

At the end of "The Story of an Hour," Louise Mallard discovers that her husband is, in fact, alive, and the feelings of freedom she had been discovering were in service of nothing. She is described at the beginning as having "heart trouble," and the shock of this discovery is what kills her; however, the interpretation of the actual shock may differ:

[Her husband] had been far from the scene of the accident, and did not even know there had been one. He stood amazed at Josephine's piercing cry; at Richards' quick motion to screen him from the view of his wife.

When the doctors came they said she had died of heart disease -- of the joy that kills.
(Chopin, "The Story of an Hour," vcu.edu)

The doctors interpret her heart failure as one of "joy," the idea being  that the strain of sudden grief and then sudden relief was simply too much for her heart. In this scenario, the joy of seeing her husband alive stopped her heart because of her weakened state.

The usual objective interpretation is that her real "joy" was taken away from her by her husband's appearance. During the hour that she thought him dead, she began to discover the possibilities of personal freedom, and of living life without her husband -- who is not a bad person -- shadowing her every move. In this interpretation, "the joy that kills" is her realization that she is not free, might never be free, and so her brief moment of true freedom and joy has been stolen from her. With her heart trouble, the powerful disappointment -- as well as the legitimate shock -- of discovering her husband alive is enough to cause heart failure.

Chopin is being ironic here.  The reader knows that Louise Mallard is actually feeling liberated by the news of her husband's passing; however, the other characters in the short story have no idea.  Her actions in that hour are considered to be ones filled with grief, not ecstasy.  Therefore, when Louise finds out that her husband is alive and well, she is so shocked (or perhaps heart broken) that she dies.  Because no one knows her true feelings, they assume that Louise has died because she is so shocked and overjoyed that her husband is actually alive.