In poetry, a refrain is usually defined as a phrase, line, or series of lines that is, or are repeated throughout the poem. By that definition, there are several possible refrains in the poem, "Caged Bird." I shall explain two phrases which might be the refrains contributing to the overall meaning of the poem.
Two phrases that are repeated throughout the poem are "free bird" and "caged bird." By repeating these phrases, Angelou emphasizes the contrasts between them. The suffering and imprisonment of the caged bird is, in other words, emphasized in juxtaposition with the joy and freedom of the free bird. For example, the free bird "leaps / on the back of the wind," and "dares to claim the sky," whereas the caged bird "stands on the grave of dreams" and his "wings are clipped and his feet are tied." A second way in which this refrain contributes to the overall meaning of the poem is by alerting the reader to the similarities that the birds share despite their different circumstances. Both birds, whether caged or free, continue to sing. The suggestion here is perhaps that each bird has a shared, inherent nature, which can be suppressed, but which can never be destroyed completely.
At the beginning of the third stanza, Angelou writes that "The caged bird sings / with a fearful trill." These lines are repeated again at the beginning of the sixth stanza. The effect of this refrain is to emphasize through repetition the fear of the caged bird, but also its hope. Indeed, although the bird's singing is "fearful," Angelou wants to emphasize the fact that it does, nonetheless, keep on singing. This alludes back to the point made above. Angelou believes that one's nature can not be completely destroyed, even by a cage that might succeed in suppressing it.