Stanza 3 Summary

Get out the microscope, because we’re going through this poem line-by-line.

I know why the caged bird sings, ah me,
When his wing is bruised and his bosom sore,--

The little birdie is singing even though it's hurt. Its wing is bruised from beating it all the time against the cage, and its bosom is sore—bad times all around.
Again, the speaker tells us that he "knows" why the caged bird is singing. He understands this bird, because he identifies with its suffering. This bird, as we can probably guess by now, isn't singing because it's happy. It's singing because it's sad.
We can also see more alliteration with the letter B here, in the words "bird," "bruised" and "bosom." "Sound Check" has you covered for all things sound-related.

When he beats his bars and he would be free;
It is not a carol of joy or glee,

This is the first time that the speaker tells us explicitly why the caged bird is singing: it wants to be "free." This sets up freedom as an important idea in the poem. This is a poem about a bird wanting, but not being able, to get free.
These lines continue the alliteration of the B words, in "beats" and "bars." B is a strong sound, and the repetition of this sound evokes the idea of the bird beating violently against the bars of its cage.
So, when we hear that really pretty song the bird sings in its cage, we shouldn't assume that it's singing because it's happy. As the speaker tells us, "It is not a carol of joy or glee." What is it then? We'll find out in the last lines of the poem…

But a prayer he sends from his heart's deep core,
But a plea, that upward to Heaven he flings—
I know why the caged bird sings!

We find out here that the bird's song isn't a song at all (though it sounds like it). It's a "prayer" and a "plea" that the bird sends to heaven.
So, what is the bird praying for? It's praying for freedom, of course. After all, in the previous lines, the speaker had told us that the bird "would be free"—that is, it wants freedom.
So when the speaker ends the poem with the words "I know why the caged bird sings!" he's emphasizing to us that he identifies with the bird's suffering, and also with its desire for freedom. This poem is as much about the speaker as it is about the bird whose suffering he describes.
A quick look back at this stanza tells us that the rhyme scheme holds with the previous stanzas. (See "Form and Meter" for more.) Like any prison, things are locked-down and orderly in this poem.