The speaker indicates that Annabel died because the angels were jealous, so they brought her a cold wind from the sea to take her up to Heaven with them.
Annabel Lee most likely died of pneumonia or a similar ailment, but the speaker has a more poetic answer. He considers Annabel an angel, and the angels in Heaven are jealous of the love that the speaker and Annabel have.
The angels, not half so happy in heaven,
Went envying her and me—
Yes!—that was the reason (as all men know,
In this kingdom by the sea)
That the wind came out of the cloud, chilling
And killing my Annabel Lee.
Poe is said to have written the poem in honor of his wife Virginia, who died of tuberculosis. Tuberculosis is an infection of the lungs that could very easily thought to be caused by a cold wind.
Virginia’s death hit Poe hard. He laments for the lost Lenore in “The Raven,” and many of his other works are dark and feature loss. By describing the angels in Heaven as taking her up with them, he can both capture the beauty of their relationship and soothe his grief with the acceptance that she is in a better place.