In saying that she would be willing to give up her life, but not herself, for her children, Edna is making a distinction between the life of the body and the life of the spirit. Edna would be willing to die for her children's well-being, but she would not be willing to live a lie for them. The workings of her spirit, being true to whom she is, is more important to Edna than life itself.
At the end of the story, the choice Edna is forced to make is, in essence, this very choice. She has, over the past months, discovered feelings and sensibilities within herself that she has always had to suppress because of the conventions of society. Because of her "awakening," Edna finds herself trapped in a life which is stifling at best. Her marriage is loveless, and as a woman of her times, she is expected to maintain an air of decorum which denies her own passions and holds her captive under the watchful and judgemental censure of her contemporaries. Having realized that to be true to her own nature, she would have to live a life entirely at odds with accepted societal and moral expectations, she considers her children, Raoul and Etienne. Edna knows that if she abandons her marriage and lives a life outside of accepted conventions, her children will suffer humiliation and ostracism, and be stigmatized by the behavior of their mother. Edna loves her children and would not hurt them in that way, but neither can she go on living a life that so denies the needs of her spirit. She chooses suicide, in effect giving her life for her children's welfare, as she cannot give her spirit, and be false to herself.