What does Luke 20:4 mean?

Jesus is talking to a group of chief priests, scribes, and elders. They are just the sort of men who would have lost money when Jesus drove out the crooked merchants from the Court of the Gentiles in the Temple Mount (Luke 19:45–46). The elders likely owned some of those stalls and gave a gratuity to the family of the high priest; the scribes are lawyers, perhaps there to make sure their clients get what they want (Luke 20:1).

These men desperately want Jesus gone (Luke 19:47–48). They ask who gave Him the authority to run out the merchants (Luke 20:2). If He says God, they can arrest Him for heresy. Instead of naming His authority, however, He names His witness: John the Baptist.

Baptism is a sign that someone wants to be known for following that teacher's message. John's baptism is best known as an agreement that one needs to "repent, for the kingdom of heaven is at hand" (Matthew 3:2). The bulk of John's message, however, is that the Lamb of God is coming, and that this figure is Jesus (John 1:29). John's purpose is to make a way so that when Jesus arrives, the people will be ready to follow Him and be saved (Luke 3:4–6).

To agree with John's baptism is to agree with his teaching that Jesus is the Messiah. If Jesus' accusers disagree, they're in social trouble. John has been dead for about three years, but his following is strong and very loyal. The crowd around them may stone them. They choose the only option that maintains their dignity: to lie and say they don't know (Luke 20:5–7).