Simone de Beauvoir begins with her question "what is a woman" because, by posing it, she can demonstrate how the concept of "woman" is problematic. Essentially, de Beauvoir's point is that "woman" is a sort of empty signifier. The term refers less to a concrete difference than to a sort of "otherness." In other words, to be a "woman" means to not be a "man."
This notion is developed carefully in the introduction to The Second Sex. Otherness is a fundamental element of human thought. Whereas other persecuted groups (e.g., Black people, Jewish people) bind together to assert their subjecthood and render their persecutors as "other," women do not. De Beauvoir writes that women "have gained only what men have been willing to grant; they have taken nothing, they have only received."
The purpose of de Beauvoir's question, then, is to suggest that "women" might actually be something—that is, that to be a woman is to assert a subjectivity and self-agency that is equal to that of men.