At the end of the novel The Bell Jar, Esther says, "I remember everything." What is the significance of this statement in the context of the society represented in the novel and Esther's experience as whole?

In The Bell Jar, Esther experiences severe depression and general dysphoria—Esther’s ideas are at odds with society's expectations and her own experience in life. Esther feels like nothing entirely lives up to the expectations she felt in early life. She has a repeated set of failures in her endeavors and is continuously faced by the limitations that society imposes on her. As a result, Esther finds it hard to continue living what she sees as a life imprisoned.
Ultimately, that conflict between what society expects and what Esther desires is what drives her into the depression that nearly ends her life. For example, when Esther wants to strengthen her ability to write creatively, her mother insists that she should learn shorthand—a marketable secretarial skill. Esther relays her feelings by saying,

The trouble was, I hated the idea of serving men in any way. I wanted to dictate my own thrilling letters. (chapter 7)

Esther’s opinion of service jobs is directly opposed to the gender norms of the 1950s. She doesn’t want to “serve” men at all, something that was far outside the norm. Her desires to be creative and live a life of independence conflict with the expectation that she will stop working, marry, and be a mother.
Esther tries to live within the expectations of society, even going so far as to get engaged. However, she experiences a break from reality when her fiancé admits that he was unfaithful and had sex with another woman, something he would not do with her.
After being rejected for a spot in a creative writing program, Esther decides that she will kill herself. After three unsuccessful attempts, she wakes up in a sanitarium.
She meets a woman named Joan, who helps her immensely. Esther eventually starts to recover; she meets a man and has sex and then ends up in the hospital after complications. Joan helps her get to the hospital, and when Esther comes back to the sanitarium, she learns that Joan hanged herself.
Esther is reeling from this information when she utters the line, “I remembered everything”—she says this in response to the suggestion that she should forget it ever happened. She says,

I remembered everything.
I remembered the cadavers and Doreen and the story of the fig tree and Marco's diamond and the sailor on the Common and Doctor Gordon's wall-eyed nurse and the broken thermometers and the Negro with his two kinds of beans and the twenty pounds I gained on insulin and the rock that bulged between sky and sea like a gray skull.
Maybe forgetfulness, like a kind of snow, should numb and cover them.
But they were part of me. They were my landscape. (chapter 20)

Despite everything she has gone through, Esther can deal with the feelings and occurrences of life that are painful. She sees this as essential to who she is as a person. The novel makes memory a constant refrain because it shows how Esther understands the world despite the contradictions, conflicts, and pain. Esther doesn’t take the doctor's advice to think that everything she has experienced is just a “bad dream”—instead, she recognizes that her experiences are a part of her and that even if they bring pain, that pain is not something she can ignore.
Esther can leave the hospital at the end of the novel, but she is still not conforming to social norms. She doesn’t want to forget. She doesn’t want to disassociate from reality—despite that being how everyone else deals with pain and difference. Her choice is to not only remember, but embrace the past—to speak to the agency she has gained at the end of the novel. She can finally make her own choices and be secure in those choices. She knows who she is and what she thinks, even if it conflicts with what people tell her to be or think.