But I laugh, And eat well, And grow strong. (lines 5-7)
Here, the speaker is making the best of a very bad situation. It would be easy to feel angry and depressed in the face of enslavement and/or oppression. But instead, the speaker in this poem <em>laughs</em>. Yes, he laughs. Why? Because he firmly believes that if he eats well and grows strong (a metaphor for coming together in solidarity as a race), he'll overcome the chains (both literal and metaphorical) that bind him. Now that's some serious ambition.
Tomorrow, I'll be at the table (lines 8-9)
A big part of being ambitious is keeping your eye on the prize: the kind of future you want. In this section, the speaker envisions a "tomorrow" in which he'll be recognized as equal to whites. He doesn't literally mean tomorrow, as in the very next day – he's talking about a slightly more distant future, maybe even a future that he won't personally get to see. But it's not himself he's worried about, really – it's all of his "darker brothers."
Nobody'll dare Say to me, "Eat in the kitchen," Then. (lines 11-14)
Ambition is also all about confidence – the ability to stand up for yourself and to command respect from other people. Here, the poem references a future in which the main character – and, by proxy, his whole race – will be treated as equal. Considering the historical situation of the poem, that's a pretty big dream. One might say a particularly <em>ambitious</em> dream.
They'll see how beautiful I am (line 16)
Like the last quote, this one defines ambition in terms of confidence. In this case, because the quote doesn't reference the future, it's more about self-confidence than about any particular kind of ambition, but it's a mindset that you do need in order to change your life. The speaker here is assertive and sure of his beauty (both inner and outer, one might assume), and this assurance makes him more capable of working towards an ideal future.
I, too, am America. (line 18)
The speaker is again talking about the present – he <em>is</em> America, right here and right now. Again, this is an instance of the kind of mindset necessary to realize (and by that we mean "to make real") a better future. Heck, this is ambition enough for a whole race. If he is America, and his whole race is America, then there's no reason why they can't demand and get the kind of treatment that they deserve.