What does John 1:37 mean?

Many commentators identify one of these two disciples as John, the author of this gospel. He is not the same person as John the Baptist, who is the one being heard (John 1:36). Ancient writings very rarely used a "first-person" narrative style. It was uncommon for a descriptive text to say, "I did this, I said that." Letters to specific people were a different form of writing, as seen in Paul's epistles. Another example is the book of Acts, which is historical, but also directed at a person (Acts 1:1).

John is also being modest by not bringing extra attention to himself. Often, John refers to himself as "the disciple whom Jesus loved" (John 13:23; 19:26; 20:20–24), or other indirect phrases (John 18:15–16). John was most likely a young teenager at this time. Though all the other disciples would be martyred for their faith, John would still be writing to fellow Christians nearly sixty years after first meeting Jesus. John was not only one of the first to follow Jesus as a disciple, but he was also the longest surviving of the apostles. He was the first of the Twelve to follow Jesus on earth, and the last of their group to follow Jesus into heaven. John's first-hand presence for many of events in Jesus' life is proven by the minute details he offers in his gospel.

The other disciple is Andrew (John 1:40).