To understand the meaning of the line "I do not ask the wounded person how he feels, I myself become the wounded person," it is important to understand it in the context of one of Walt Whitman's most important themes in "Song of Myself." Whitman celebrates his uniqueness, individuality, and importance in a cosmic sense. He also expresses his identification and empathy with others, and his unity with the universe and with nature.
This quote concerns the deep empathy he feels for his fellow humans. He feels this empathy so strongly that he becomes the person that he empathizes with. He shares their pain and exults in their joy. In the poem, his empathy for humankind includes everyone, no matter how different one person is from another. That's why in the beginning of the poem he writes that "every atom belonging to me as good belongs to you," and at the end of the poem, in part 51, he writes, "I am large, I contain multitudes."
Whitman gives clues of this total inclusivity and universality of existence throughout "Song of Myself." For instance, he opens part 16 by declaring, "I am of old and young, of the foolish as much as the wise" and then goes on to delineate examples of wildly diverse types of people from different locations and occupations and insists that he is all of them. In part 24 he proclaims that he speaks with the voices of many different people and says that "whoever degrades another degrades me, and whatever is done or said returns at last to me." He is again expressing his empathy for others.
In part 33, Whitman takes his empathetic awareness to the ultimate level by declaring that he not only sympathizes with people undergoing trauma, but he becomes these people. As examples he writes that he is a wife screaming at the sight of her drowned husband, a runaway slave chased by dogs and riders, a fireman crushed as walls tumble onto him, and a wounded person in agony. All of these examples serve to highlight Whitman's, and by inference the reader's, unity with all of humankind.