Wollstonecraft mentions on several occasions that women exist inside symbolic cages because of the way male-dominated society has treated them.
But instead of trying to break out, these women focus on all of the things society has taught them to value, like looking pretty like birds: "Confined then in cages like the feathered race, they have nothing to do but to plume themselves, and stalk with mock majesty from perch to perch" (4.11).
The image of the birdcage is a pretty heavily used one: think of Maya Angelou's famous poem. Wollstonecraft uses it really skillfully, however: birds are thought to be dull, vaguely entertaining and pretty animals. Women are thought to be dull, vaguely entertaining and pretty beings.
The cage is symbolic of the patriarchy. Not only is it enslaving, but it also starts to feel a bit like home after a while. Domesticated birds are reluctant to leave their cages: the cages are all they know, and they're ill equipped to function in the wild. Women in Wollstonecraft's time were reluctant to leave the fold of the patriarchy, in large part because their lack of education made it almost impossible for them to find gainful work.