And if you are not needed by your brother men, there is no reason for you to burden the earth with your bodies. (1.20)
Equality 7-2521 expresses the basic principle of his society here. It holds that human individuals are a burden and have no reason to live unless they can be of use to others and the greater good in some way. This ultimately makes each individual feel guilty for being alive. The message is that individuals need some higher reason (serving others) to live aside from themselves, or else their existence is unjustified. It is exactly this idea that Ayn Rand militantly rejects as false morality, and Equality 7-2521 himself will realize this by the end of Anthem.
We shall never leave this house," we said, "nor let it be taken from us. This is our home and the end of our journey. This is your house, Golden One, and ours, and it belongs to no other men whatever as far as the earth may stretch. We shall not share it with others, as we share not our joy with them, nor our love, nor our hunger. So be it to the end of our days." (10.15)
Equality 7-2521 has invented the concept of private property here. He's claimed a "home" for himself and his beloved, and doesn't intend to share it with anyone else. It belongs to him as an individual, and to him alone, just like his hunger and the love he feels for certain other individuals. Having a physical space and objects that belong only to oneself is seen as an important dimension of freedom by Rand, who made private property central to her political philosophy. She asserted that private property is crucial for separating oneself from the potentially oppressive presence of other people. As Equality 7-2521 will say later on, human freedom is freedom from other people.
I wished to know the meaning of things. I am the meaning. (11.4)
Equality 7-2521 has discovered the word "I," and with it the central truth of Randian egoism. Before he wanted to know what things in the world "meant," and what the meaning of his own life was. But he's now found out no such meaning exists. His life has the meaning he gives it. And it's not Collective 0-0009 humankind that gives it this meaning – it's him, as a unique individual.
I wished to find a warrant for being. I need no warrant for being, and no word of sanction upon my being. I am the warrant and the sanction. (11.4)
Equality 7-2521 has now explicitly rejected the principle of his society that he stated in the first quote. He no longer needs a justification for being alive. He himself, and his happiness, is his own reason for being alive. This is tremendously liberating, because it means he no longer needs to feel guilty about being alive, or pursuing his happiness. He decides that is the only moral thing to do.
It is my mind which thinks, and the judgment of my mind is the only searchlight that can find the truth. (11.5)
Equality 7-2521 has made a bit of a venture here into Rand's philosophy of how real knowledge is arrived at (which she developed much more in her later works). It is individuals, who learn about the world by studying it carefully and seeing what it is like. Society, on the other hand, has a tendency to conceal the way the world is, because it tries to force everyone to understand the world the way it thinks it should be understood. In Rand's philosophy, individuals are the real bearers of truth.
And my happiness needs no higher aim to vindicate it. My happiness is not the means to any end. It is the end. It is its own goal. It is its own purpose. (11.8)
Individual happiness, as Equality 7-2521 says here, is the highest goal of life. It is an end in itself. There is no larger moral purpose to life, other than individual happiness. This is the central ethical idea of Rand's egoism, and it's radical. It's opposed to any philosophy that holds that individuals can or must serve a higher purpose than their own happiness. No such obligation exists.
I do not surrender my treasures, nor do I share them. The fortune of my spirit is not to be blown into coins of brass and flung to the winds as alms for the poor of the spirit. I guard my treasures: my thought, my will, my freedom. And the greatest of these is freedom. (11.11)
Equality 7-2521 refuses to share here. He has no obligation to share with others, he says. Rand thinks you need to unlearn what may have learned as a kid. The highest goal in life is your own happiness, and so whatever is essential to that life and happiness you have to protect from the others who will try to take it away. Also, Rand may be alluding to (and rewriting) Biblical scripture here (see "Symbolism, Imagery, Allegory").
I am neither foe nor friend to my brothers, but such as each of them shall deserve of me. And to earn my love, my brothers must do more than to have been born. I do not grant my love without reason, nor to any chance passer-by who may wish to claim it. I honor men with my love. But honor is a thing to be earned. (11.13)
Equality 7-2521 tells us that he does not have any obligation to serve "mankind," or to love it. He can only love people who he respects and cares for, and that doesn't just come automatically. Other individuals earn his love and respect, on the basis of their own individual qualities and their interactions with him. The people he likes may be an integral part of his own happiness (like Liberty 5-3000 is). This constitutes another key idea of Randian egoism.
For the word "We" must never be spoken, save by one's choice and as a second thought. This word must never be placed first within man's soul, else it becomes a monster, the root of all the evils on earth, the root of man's torture by men, and of an unspeakable lie. (11.15)
What's the root of all evil according to Equality 7-2521? The first-person plural. The moment society or any group larger than the individual is made more important than the individual, the individual's freedom is threatened. And that's both not so great for individuals, and immoral (according to Rand's philosophy, the highest moral goal of each individual life is happiness). It's also bad for society, in the long run, as Equality 7-2521's society shows clearly.
And now I see the face of god, and I raise this god over the earth, this god whom men have sought since men came into being, this god who will grant them joy and peace and pride. (11.20)
Equality 7-2521 worships his own ego, although it's important to clarify here. First, Equality 7-2521 isn't just worshipping his own ego, he's worshipping the notion of "ego" in general. And by "ego" he means the idea that human individuality and selfhood is what gives the world meaning. (Rand later identified this with human "reason.") Equality 7-2521 has to worship that capacity in himself, of course, but he also worships it more generally. Second, we usually tend to think of selfishness as something bad and use "ego" as a nasty word, so when we say something like "he worships his ego." That's because we think that you're supposed to worship something higher than the ego. Rand believes that the ego gives meaning to life and, and as such, is worthy of worship. It also helps that there's nothing else to worship with which it might compete