"After these things..." refer to the Passover and Nicodemus's visit. It is important to note that Jesus was gathering quite a crowd here, and that there was quite a religious sentiment going on in the Judean region, especially in Jerusalem. Also note that Jesus' disciples were baptizing. Referring back to the section in the John posts on baptizing we see that it was quite a common practice, however, its origins are not clear, and what Jesus' disciples were baptizing the people for is also not mentioned. It might be that while they were baptizing, Jesus was also teaching, since it is mentioned that He Himself did not baptize. This also tells us that Jesus' disciples at this time might have some religious status and recognition, being associated with a popular teacher, Jesus.
In any case, there arose a debate. How often, when there is a large crowd of believers, even religious leaders, there will also be a debate. And note that it was a purification, a debate about rituals probably between John's disciples and the Jews (probably the Pharisees), a dispute about who is doing what and who is right about it. Perhaps we can get the sense here that while the Pharisees were in legal religious authority, there were bands of teachers who might have disagreed with them, and there was constant voicing of religious ideas. Can we map this on to the way churches and religious organizations behave now?
But note even more surprising is that while the dispute was about purification, what John's disciples asked John about was nothing about purification but they were concerned about Jesus having more followers than John! Perhaps they were not really concerned about purification after all, perhaps it was bothering them that this thing was happening, and the unrest made them choose a topic just to vent? Again, do we do that ourselves? This is a call to re-examine our motives, no matter how admirable, we may have the best of intentions to dispute about religious motives and practices, but is that really our concern, and even more important, is that really what matters? Did we miss the point about what Jesus is doing?
Examine John's response deeply, because it seems in contrast to his disciples' and perhaps may be what John the author is putting as an example for us to follow in response to the Christ.
"He must increase, but I must decrease".
We are but witnesses to a more glorious Light. Always remember that. Be humble. Especially in the presence of One who comes from above and testifies to that which He has seen and heard. This statement constitute another testimony from John the Baptist as quoted by John the author, another testimony about the Christ. And Christ's testimony, through His actions, about Himself.