Auden wrote "The Unknown Citizen" in 1939 after he moved to The United States. It was published early in 1940. At this point, World War II had begun in Europe, but the US would not enter the war for almost two more years. (When the poem mentions "the War," it is referring to World War I.)
The poem critiques the way modern society instrumentalizes human beings. To instrumentalize a person is to use them merely to extract as much value or profit from them as possible. It is considered unethical, for instance, to have a second child so that that child's bone marrow can be extracted to treat an illness in the first child. A person should not be birthed simply to be used.
In this poem, however, the state finds in the unknown citizen the model citizen because he is utterly instrumentalized. He completely conforms and does everything he is supposed to do, without deviation. He comes to work regularly and without complaint, so society is able to extract maximum value from his labor. He also does exactly what he is supposed to do with his leisure time: he was "popular with his mates and liked a drink." In other words, he doesn't do anything to create headaches or require the state to expend extra resources "fixing" him. In fact, by drinking, he drowns any disquiet he might feel. He also does his part in having a family to provide more instruments for the state to use: in fact, he has five children. Finally, not only does he work efficiently, he also consumes, keeping the machinery of capitalism profitable. His lack of a name symbolizes that he is valuable to the state only for what he does for it, not for himself as a unique being.
Auden ends the poem by asking the more abstract question of whether such an individual is free or happy and having the state dismiss such musings as "absurd." Auden wants to convey that it is dehumanizing for people to be treated as if they are little more than machines to be programmed and worked until they wear out.