When Edna learns finally to swim she gains great confidence in herself, her own 'powers' and 'strength':
A feeling of exultation overtook her, as if some power of significant import had been given her soul. She grew daring and reckless, overestimating her strength. She wanted to swim far out, where no woman had swum before. (chapter 10)
The sea is an important symbol in this story. It represents individual liberation. When Edna learns to swim she learns to rely on herself. This is the 'awakening' of the title: her growing awareness of her own independent identity, where she does not need other relationships, even those of her family. This is symbolized in the way that she shuns the company of other swimmers.
Thereafter, Edna grows ever increasingly independent in her ways. This is seen almost immediately, when she returns home and lies languorously out in the hammock, refusing to listen to her husband's entreaties to come in, even although it is very late at night. She begins to alter her lifestyle completely, rejecting her former acquaintances, and going out on a Tuesday, the day of the week when she is expected to stay at home to receive visitors. Such behaviour on her part utterly perplexes her husband, who begins to wonder if she is suffering from a mental breakdown. She also begins a dalliance with Robert, although, in the end, even he does not satisfy her completely. What she really desires to be alone.
In short, Edna begins to shun society, to become entirely her own person. The story focuses not just on her own individual liberation, though, but her awakening as a woman, her repudiation of the domestic role that a woman was traditionally expected to occupy, looking after husband and children, entertaining the neighbours and being the perfect hostess, and so on. The feminist significance of her rebellion is symbolised in the quote above: 'She wanted to swim far out, where no woman had swum before'. In other words, she wants to break the rules for women, to defy society's expectations of her as a wife and mother.
The quote above also hints at the danger of what she is doing; it is said she 'overestimates' her own strength. In the end, she is not really able to cope, and drowns herself - although this could also be figured as her ultimate escape from the demands of society.