"Truly, truly, I say to you, a servant is not greater than his master; nor is he who is sent greater than he who sends him" (John 13:16 RSV).
I am a student of Truth at Unity Church in South Africa.
As you noted, Jesus provides his own commentary on this verse a little later, at John 15:20 (RSV): "Remember the word that I said to you, 'A servant is not greater than his master.' If they persecuted me, they will persecute you; if they kept my word, they will keep yours also."
Jesus Christ is speaking at the Last Supper as he nears the conclusion of his earthly ministry. He is speaking, not merely as Jesus of Nazareth, but as Jesus the Christ, fully expressing his spiritual essence as he calls his followers to learn to express theirs. It won't be easy—there will be resistance to them as there is to him. But there is always hope. There will be those who believe in his teachings and, as a result, believe in their work. In other words, what they believe about the teacher will define what they believe about the students.