This constant refrain is meant to convey the powerful message that whatever happens to African-Americans, however many times they're subjected to racism, prejudice, and bigotry, they will never be kept down. They will always rise up and assert their identity in the face of those who would keep them in a state of permanent subjection.
"I rise" is intended as a message of hope and inspiration, an injunction to dig deep in the face of oppression, to stand tall and not be ashamed of one's identity. Most people are familiar with the expression of picking yourself up, dusting yourself off, and starting over again as a response to life's numerous setbacks. The same principle is in operation here.
The speaker seems to suggest that African-Americans will always be subjected to oppression of one sort or another. That being the case, there's only one way for African-Americans to maintain their dignity and sense of self-worth, and that's by rising up every time society tries to keep them down.