There is a big difference between power and authority.  Power underlies the ability to do something, but authority determines the right to do it.  Jesus had demonstrated many 'works of power', and even the religious leaders, who wanted to get rid of Jesus, could not deny the powerful miracles . So, they asked what the basis of His authority was to do them.  These people were the religious authority in Israel, and Jesus was fundamentally threatening their powerbase.  Even Pilate knew that their hatred of Jesus was because they envied His superior power (Mark 15:10).

They reasoned that if Jesus claimed it came from Father God, they would accuse Him of blasphemy; and if He should say it came from demons, they would accuse Him of sorcery (John 8:48-51): both would have been sufficient grounds to put Him to death ... which was their aim.  But as with every scheming plot against Jesus, He knew what they were thinking (Luke 6:8), and never fell into any of the traps they set.
 
So, Jesus responded by asking them the same question about John the Baptist's authority. The religious authorities had been glad when he was beheaded (Mark 6:17-29), because his preaching humiliated them (Matthew 3:7-10).  However they dared not move against Jesus in public.  Like John the Baptist, Jesus was the popular hero.  If the chief priests even spoke against Jesus they would risk a riot and the possibility of the Romans removing their right to control the populace (John 11:48).  So, they said that they did not know the source of his authority; to which Jesus responded that He would not answer their question either.
 
Nobody ever caught Jesus out.  He always had the right answer and would never be trapped by those who refused to submit to Him.  The reason is that His authority was the same as Father God (John 8:28).  Colossians 2:9 puts it like this, "For in Christ all the fullness of the Deity lives in bodily form ..."   That is the reason why billions of people around the world trust Jesus today; because it is clear that His authority is higher than any on earth and equal to that of Father God.  All His words and works prove that to be true.  Of course, many who have no desire to call Him Lord will try to ridicule His authority (1 John 4:1-3); but the proof of His Divine authority will come when He returns in power to judge the world (Acts 17:31).  However, wise people will not wait until that day to submit to His Lordship!