How did Sai Baba explain his philosophy of oneness with all living beings to the lady in the story?

Sai Baba explained his oneness with all life by connecting his own state of being to the lady's recent actions. As detailed in Chapter 9, when the lady was confused about how she could have served him, Baba replied that by her feeding a hungry dog, he himself became satisfied and was still belching. He explicitly stated, "that dog which you saw suddenly, I have oneness with him," and similarly with a pig. He generalized this principle by saying, "Sometimes I am a dog, sometimes a pig; sometimes I am a cow, sometimes a cat... in such forms I roam." He concluded by teaching that true worship is to renounce the sense of difference and see him in all beings.