This paragraph explains how and why men think and act important. Formal authority here means laws, policies, and social positions that entitle men to think and act important. "back them up for as long as they still serve" basically means men will have aforementioned official entitlement long as they still hold their social position. For instance, we believe it is normal and proper for Barack Obama to think and act important as long as he is the president right? However, when he stepped down the president position, he can still think and act important, but he is not the president anymore, so he loses the "formal authority". You get that?