What does the nationwide "Quit India" movement symbolize when Gandhi advocated for it?
I would say that the highest and most relevant form of symbolism of Gandhi's advocacy of the "Quit India" Congress ideal is that it was one of the last moments that the British could claim power over India. In his "Quit India" speech, Gandhi makes it clear that if all Indians sacrifice in the "Quit India" movement, the British will have little choice but to surrender India to Indians:
We cannot evoke the true spirit of sacrifice and velour, so long as we are not free. I know the British Government will not be able to withhold freedom from us, when we have made enough self-sacrifice. We must, therefore, purge ourselves of hatred.
Gandhi's advocacy of the "Quit India" ideal was that if Indians shed their distinctions amongst one another, antagonisms that the British were able to use to keep Indians divided from one another, the British would not be able to sustain their control over half a million Indians. In this, the "Quit India" advocacy became another forum for Gandhi to make clear that spiritual and moral transformation is the basis for which all political change can be recognized. Gandhi saw that his advocacy for this ideal would be able to deliver political change, but having done so on an elevated moral and spiritual consciousness. The symbolism of India as a nation towards the greater good of ethical change is where Gandhi realizes political change can happen.