This utterance is significant because it marks what could be a turning point in Caliban's life.
Up till now, he's been seeking out the women he thinks will change him, because he wants them to turn him into whatever they want him to be. (As has been discussed in other answers already, this represents Caliban's desire to be part of 'white' society, to leave his island roots behind and become whatever the "golden girls" want him to be.) We see this from the very start of the story, when he tries to get his current girlfriend to kiss him and ask him to be her frog, up until near the end:
You risk a glance at the woman you’ve dragged into this, the golden girl. She’s standing now, a look of interest and curiosity on her face. “This is all your fault,” you say to her. “If you had kissed me, told me what you wanted me to be, she and Ariel couldn’t have found us.”
“And you,” says Sycorax, pointing at you with a suckered tentacle, “you need to stop bringing me the fallouts from your sorry love life.”
“I can’t help it, Mama,” you say. “That’s how women see me.”
Immediately after this exchange, Caliban's mother Sycorax loses her temper and begins to transform him into the way she sees him: a little boy. This is the opposite of what he wants: he's seeking to leave behind his roots (and by extension his family, whom he seems to avoid as much as possible).
This is when his girlfriend steps in and begins to change him, but it doesn't go exactly as he'd hoped.
- First she turns him into a sunflower, which seems to be more along the lines of what his girlfriends usually do to him. He "stand[s] rooted, waiting for her direction" - once again, looking to the white woman to tell him what to be.
- Then she turns him into a baby waiting for his mother. This isn't the treatment he normally gets from his girlfriends (perhaps because none of the previous ones had met his family before?) - his reaction this time is "Oh, please, end this". This girl isn't what he expected: she's made of tougher stuff, able to stand up to his family and even to sympathise with his mother.
- Next she turns him into a man again, sexually aroused and reaching for her. He's back into familiar territory now: "So familiar, the change that wreaks on you." But the final change is yet to come.
After all this, she asks him the question which, it seems, none of his "golden girls" has ever asked before:
“Who do you think you are?”
Caliban is shocked. He looks to her for an answer, but her eyes are blank: she doesn't see him as anything any more; she wants to know what he sees himself as. This is, in some sense, a level of respect he's not used to. When he's been with white women before, they've always transformed him into whatever they want him to be. This one wants him to be himself. In fact, her parting remarks force him to be himself if he wants to see her again:
She looks at you, and her eyes are empty, open, friendly. You don’t know what to make of them. “Um,” she says, “maybe you can give me a call sometime.” She starts walking away. Turns back. “It’s not a brush-off; I mean it. But only call when you can tell me who you really are. Who you think you’re going to become.”
When white women look at him, he's used to seeing their idea of what he's like, and changing himself to adapt to it - whether that's the racial stereotype seen by Miranda long ago, or the dehumanising frogs and dogs whose offspring he's brought to his mother. Now, he's found one who respects him enough to let him be himself ... if he can figure out who that really is.
The story leaves open the question of what he actually does after that. Maybe he's about to embark on a journey of self-discovery, reclaim his identity and self-respect, and find the girl again. His use of the word "I", when he's always been a second-person narrator, gives us hope for this outcome. But the story ends there, letting us exercise our own imaginations to speculate about what happens next.
This personal story of a man and a family reacting to the effects of colonialism could also be interpreted as an allegory for post-colonialism itself. (Disclaimer: this is a topic I don't know much about. I'm trying my best, but if I say something wrong or insensitive, no offence is intended. Please set me right if I'm grossly out of line due to ignorance.)
Jamaica, and many other countries in the world, have experienced colonial rule: they started off inhabited by indigenous peoples with their own cultures; then European colonists arrived and suppressed that culture with theirs; finally the countries became independent, but still needed time to find their feet and define themselves. I don't know about Jamaica specifically, but in many places the original native culture has been so suppressed by colonists that it's hard to reconstruct in the post-colonial era. Perhaps Caliban represents those who cling to the colonial way of life, while his mother and sister represent those who prefer the old ways. The final sentence of the story indicates a third possibility: to redefine the culture in a new way, acknowledging their colonial history without being defined by it. It's possible for a black island man to be together with a white woman without being defined by her perception of him. It's possible for him to be true to himself without betraying his roots or despising the descendants of the colonial oppressors.