The "outward life" describes the social persona—the mask individuals assume when out in the world among other people. This persona is taken on so an individual can comfortably engage in society without seeming like a freak or an outsider. The "inward life" is the person behind the social mask. This inner self may or may not agree with the rules that come with engaging in the public social world, hence the verb "questions."
In The Awakening, the protagonist, Edna, is someone whose outer and inner identities are at odds. While she appears to be a dutiful wife and mother, in truth she gains no satisfaction from either of these roles, even though society expects her to do so. She does not love her husband, even if she is fond of him in a casual way. She is not mad with devotion toward her two children either. At one point, the narrator mentions that, when the boys were staying with their grandmother for a long period, Edna rarely missed them:
Their absence was a sort of relief, though she did...
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