What does John 13:23 mean?

Ancient literature often wrote in third person, which is the description of participants as "he, she, and they," rather than as "I" or "you." That was true even when someone wrote about first-hand experiences. In this case, the disciple "whom Jesus loved" is John, himself. This identity can be pieced together by John's other references to this person (John 19:26; 21:7), compared to those same events as described in other Scriptures (John 21:2; 21:20–21; Matthew 4:21; Matthew 17:1; Mark 5:37; Luke 8:51). John's use of the term is meant, in part, to imply his especially close friendship with Jesus. In a literary sense, it's also a sign of humility, rather than to say, "it was me—I did that!"

Dinner customs of that era involved eating in a reclined position. People would lay on their stomach or left side and eat with their right hand. The disciple spoken of here will "lean back" against Jesus to ask Him a question, implying that he's to Jesus' right (John 13:25). That detail helps make sense of what happens next: Jesus will refer to a sign indicating which person is the traitor, but it seems this disciple does not see it. That's easier to understand when one considers that he's reclining with his back toward Jesus.