I think we can safely assume that this argument was about baptism. This section starts off talking about baptism and immediately after we hear of the argument it goes back to baptism. So my guess is a Jewish person, probably a Pharisee like Nicodemus is trying to understand why Jesus and John are baptizing Jews. It also should be mentioned that Jesus was not baptizing himself but His disciples were (John 4:2).
So after this discussion John’s students go back to him and they are worried. They are afraid that this man Jesus is going to get a bigger following than John. I think today’s application on this is the church and it’s constant concern for other churches getting more members. I have heard church members talk about not wanting to work with a church on something because they are afraid they may lose some members. What we must realize is that God sees one church and one denomination. It should thrill us when we hear about a church growing. The angels are throwing a party so why aren’t we(Luke 15:10).
If you read the rest of this section with that in mind it will help you see that each of us are just a part of announcing that Christ has come. We should seek to be humbled and made small as we proclaim this news to the lost. We should be thrilled for the groom (Christ) when we hear of another coming to Him. Be it an individual or a growing church.