"... you can't take her with you, wherever it is you're planning to go, when you're making your clever speeches, trying to whip yourselves up a following."
-- Aberforth Dumbledore, recalling what he said to his brother just before the duel began
Angry that his brother is neglecting their fragile sister for his new friend, Aberforth confronts Albus and Grindelwald about their plans to leave Godric’s Hollow and make their mark on the world. Grindelwald tells Aberforth he just doesn’t understand, and then uses the Cruciatus Curse on the younger boy. Then a duel break out between the three wizards. The duel overwhelms Ariana, and she reacts in some way that causes her to be caught in the crossfire. It is unknown who cast the spell that killed her.
This probably happened in late August since Aberforth says that it happened shortly before he returned to Hogwarts, which usually begins the new term on September first.
The reader learns of this event when Aberforth Dumbledore recounts it to Harry, Ron, and Hermione. When Aberforth finishes telling the story, he suggests that his brother Albus was glad to be free of the sister for whom he had been obliged to care. Harry disputes this, recalling the way Professor Dumbledore had pleaded for someone to hurt him instead of "them" when delirious from the potion he drank to retrieve the locket in the cave. Harry says he is certain that Dumbledore's words were about the duel that killed his sister.