What argument is she making about the “objectification” of women?

Woolf argues that, historically, women have lacked the same opportunities that men have enjoyed. She is, in particular, making this case in terms of women writers and how women had been represented in literature. The notion of "room" or "a room" is literal and figurative. Women need a room, a space to write and be published. They need to be paid; thus, they need the opportunity for that employment. Room needs to be made in the profession of writing and room needs to be made in the canon of literature (in terms of allowing more female writers and in terms of more diverse representations of women in literature). Woolf adds that, historically, women in literature have been represented primarily as the objects of men's affections. Being represented mainly as these "objects" severely limits the ways women can be represented.
In the section “Shakespeare's Sister,” Woolf imagines that Shakespeare had a sister (Judith) who had the same gifts that he did. Woolf supposes that Judith would not have had Bill's (Shakespeare's) access to education. Judith would also not be permitted to act in plays (all roles at this time were played by males). There is no "room" for Judith. Woolf assumes that faced with a life of being denied the ability to develop and share her genius, Judith would be miserable and eventually commit suicide. Judith would have been treated as an object: a daughter to be married off with no consideration for what she wants out of life. Judith is an object to be exchanged. Meanwhile, her brother, Will Shakespeare, gets to thrive in his genius and make the most of his life. Woolf thinks that women (like Judith) should have the same opportunities as men. Denied the ability to be an author (of literature and of her own life), Judith is an object. If room was made for the Judiths of the world to write and have equal opportunity, they would shift from objects to authors. The author has authority over her own life. The object does not.
The same reasoning goes for how women are represented in literature. In the section on Chloe and Olivia, Woolf writes about how women in literature are mostly represented by their relation to men. (This is called the "male gaze"—that women are only seen/represented through men's perspectives.) Up until Jane Austen's time, women in literature were mainly objects of men's desires. Woolf wonders what Antony and Cleopatra would have been like if Cleopatra and Octavia might have been friends or more. Male characters in literature had always been described in great and various detail. This had not been the case with women. Woolf adds, “Suppose, for instance, that men were only represented in literature as the lovers of women, and were never the friends of men, soldiers, thinkers, dreamers . . . “ Woolf simply wants the same for women—to be more than objects of men's affections.