In this poem, the speaker (Maya) is addressing her oppressors (members of the current and traditionally white oppressive culture). She acknowledges how she (and her people) has been wronged, yet asserts her determination to live life to the fullest: for her own sake and in homage to her ancestors. She dares her oppressors to try to keep her down and dares them to be upset at her determination.
The first stanza is important in terms of the oppression she's addressing.
You may write me down in history
With your bitter twisted lies,
You may trod me in the very dirt
But still, like the dust, I rise.
The oppression and racist thinking is literally written in the history books and also "inscribed" in the consciousness of America. This shows how systemic racism is a part of the nation's literature and its culture. It is therefore fitting and effective to attack this systemic written racism and culturally inscribed racism with writing itself (the poem): to "right" (correct) the historically racist writing and thinking, so to speak.
In the fourth stanza, she asks if they (oppressors) wanted to see her broken; she is determined not to let that oppression affect her life. To be sure, she is raging against the oppression, but she won't let it bring her down: "Still I rise."
She also addresses the way she is viewed (and literally "looked at"), with hate, by the oppressors, "You may shoot me with your words / You may cut me with your eyes."
Her difficult circumstances go back generations. Strides may have been made in the progress of civil rights, but the pain goes a long way back and is by no means gone yet:
Out of the huts of history's shame
Up from a past that's rooted in pain
She finishes the poem by acknowledging the historically painful past of her people but ends with hope and again on the eternally optimistic refrain:
I am the dream and the hope of the slave.