What is the ending trying to say? Like who is "her" and "his"? Who is the person of which they do not know who will be born into a curse of slavery becuase I am really confused.

The last thing to go was a tiny bundle of letters; innocent little scribblings that Désirée had sent to him during the days of their espousal. There was the remnant of one back in the drawer from which he took them. But it was not Désirée’s; it was part of an old letter from his mother to his father. He read it. She was thanking God for the blessing of her husband’s love:—

“But, above all,” she wrote, “night and day, I thank the good God for having so arranged our lives that our dear Armand will never know that his mother, who adores him, belongs to the race that is cursed with the brand of slavery.”

"She" is Armand's mother. "Him" is Armand. Armand's mother was a slave and he "belongs to the race that is cursed with the brand of slavery." From this, we learn that Armand never knew his mother was black, and that Desiree's baby had the complexion of Armand's mother..... not Desiree.