As with the last two lines of "In Memory of W. B. Yeats," most of the other parts of the poem aren't rosy. In the first stanza, Auden presents a harsh scene. Yeats died in January 1939, so it's the "dead of winter" and a "dark cold day." To drive home the austerity and frosty tone, Auden repeats the image of the "dark cold day" at the end of the first section.
The second section continues to evoke the malaise of the world. As many of Yeats's poems touched on Ireland and its political conflicts with England, Auden, addressing the departed Yeats, speculates, "Mad Ireland hurt you into poetry."
In the third and final section, Auden appears to link the violence that Yeats witnessed with the violence of World War II, which began in September 1939. "In the nightmare of the dark / All the dogs of Europe bark," says Auden. In other words, Europe is as violent as it ever was.
According to Auden, reading Yeats can counter the "intellectual disgrace" that initiates war. Yeats is dead, but his poetry lives on. Through his poetry, he can "persuade" people to "rejoice" instead of fight.
The notion of hope seems to be the central significance of the poem's last two lines. Humans are not in a good place. They live in a "prison." Yet Yeats's body of work can "teach the free man how to praise." With the last two lines, Auden cements his claim that poetry like Yeats's can help people overcome their narrow, violent hatred and express positive emotions.