We talk about “1960s pop anthems,” but what if it were possible to take soul, folk, and rock opuses from the era of pot-and-protest and reinterpret them in a completely personal and intimate way? That’s what the highly individual singer-multi-instrumentalist Nellie McKay is doing this weekend at 54 Below and on her new album, “My Weekly Reader.” Throughout, she takes songs that were anthemic or dogmatic and reweaves them into something warm and vulnerable—stories about human beings, rather than abstract causes. Most of us have grown up accepting these lyrics without questioning their deeper meanings, but Ms. McKay extracts their inner truths by illuminating their foundations: the barn dance vaudeville of “Fixin’ to Die Rag,” the delicate balance of defiance and compassion at the heart of “Don’t Let the Sun Catch You Crying.”