Anthem- "I owe nothing to my brothers, nor do I gather debts from them." What does this quote mean?

This quote relates to freedom and independence. Equality owes nothing to anyone, nor is owed anything by another. He is no longer thinking in terms of we, but rather in terms of himself. Life is no longer to be lived for the collective good, but rather lived in an independent way, according to Equality's own wants and needs.