That was a direct affront to those who were in charge of the Temple and the Sanhedrin. So their question to Him is: "Who gave you the authority to do these things?"
This was a crucial incident because the Sanhedrin was directly challenging Him. Their response shows us a situation we want to avoid happening in our life. The term "baptism" is simply a way of indicating John's whole ministry. The question was, "By what authority?" Jesus had no authority from the Temple; He had no authority from any priestly school; He had no authority from the Sanhedrin for what He was doing.
But His question on the return put them right on the horns of a dilemma. If they said John's authority was from God that would also validate Christ because John said Christ was the Messiah. If they said John's authority was not from God then they had to face the anger of the people, because the people looked upon John as a prophet from God.
So they dodged the question by claiming ignorance. They dodged the truth by claiming ignorance. Their attitude toward truth led them to expediency rather than principle. They became pragmatists. They did what was practical in the situation rather than following through with the ideal, which would have been to answer honestly. They did what they felt was safe rather than truthful and their pride of position led them into this evasion (because that is what it was).