What lies below is certainly not the only way to interpret this scripture, but it is one way I find extremely compelling, and to my knowledge, provides a reasonable historical understanding.
This passage in scripture is built on a long foundation of culture and history, which is largely lost on a modern audience.
First, a reminder about the immediately apparent context: Jesus had just entered the temple and driven out the merchants.
Also remember the larger context, that the Jews are living under occupation, and they resent having to pay taxes to a foreign ruling power, and they are eagerly awaiting a messiah to come and rescue them from this oppression--some are expecting a military leader to lead a rebellion against Rome. Practically all Jews are expecting some sort of political leader (there wasn't a clean distinction between politics and religion then, as there often is in our minds today).
So when Jesus was asked about paying taxes to Caesar, it was a loaded question, not really about taxes, per se, but inviting Jesus to take sides on the issue of open rebellion.
But that's not enough to fully demonstrate why Jesus' questioners were amazed. As N.T. Write explains briefly in Jesus and the Victory of God, Jesus' words "Give to Caesar what belongs to Caesar, and to God what belongs to God" mirrors a revolutionary phrase, which would have been well known in contemporary Jewish culture of the time.
A hundred years earlier, during the Maccabaean rebellion, Mattathais made a famous death-bed speech, which concluded with "Pay back the Gentiles in full, and obey the commands of the law"--a command to, according to the law of God, rebel against, attack, and destroy the pagans.
Jesus mirrored this popular "call to arms" to say, in essence, "Give Caesar what he deserves." But he said it in the context of paying taxes. Was Jesus demanding that Jews pay taxes? Was he demanding that they rebel against Caesar?
As Wright concludes, He had done neither. He had done both.
This is why they were amazed... and probably incredibly confused.
Here I quote the relevant section of Jesus and the Victory of God, for greater context. I've still edited out a few bits, so as not to make it any longer than absolutely necessary.
The question of tax-money was not simply a trick, designed to frame a charge against Jesus. Jesus' reply, likewise, should not be read simply as a cunning avoidance of the question, still less as a way of shifting the discussion from 'politics' to 'piety'. Tax and Temple, Caesar and God, are the subject-matter.
'We know that you ... show no respect of persons.' This preamble assumes that Jesus is not afraid of Caesar, and so will give the revolutionary answer if that is what he believes. That, at least, is what the questioners assume the bystanders will be thinking.
Jesus' pithy reply encapsulates the larger issues of his own doubly revolutionary kingdom-agenda. He began by requesting one of the relevant coins. This took the initiative away from his questioners, forcing them to reveal their own hand first. The coin bore an image and superscription which were, from a strict Jewish point of view, blasphemous. The image was prohibited (even the cynical Antipas, as we saw, had stopped short of using an image of himself on his coins), and the superscription proclaimed Caesar in divine terms, specifically as the son of a god. Jesus' questioners were thus themselves already heavily compromised by possessing such an object.
Jesus then responded with the famous two-line aphorism: give to Caesar what belongs to Caesar, and to God what belongs to God. This has often been taken to imply a neat division of loyalties: state and church, Caesar and God, held in delicate tension. Alternately, the saying has been read as a wry, ironic comment, or, indeed, as a direct challenge to zealotry. But there is reason to suppose that both parts of the aphorism are more subtle, and more closely linked to the issues of Temple, Messiah, and Jesus' whole kingdom-announcement, than those options allow.
Jesus' hearers would have been expecting some kind of signal that he was indeed in favor of revolution. It might be cryptic, but in many political situations coded statements are all that one can offer. I suggest that Jesus deliberately framed his answer in terms that could be heard as just a coded statement, with which he neatly refused the either/or that had been put to him and pointed to his own kingdom-agenda as the radical alternative.
The initial clue is found in a passage probably familiar to Jesus and his audience, if only through the regular celebrations of the Maccabaean heroes at the annual festival of Hanukkah. As the old revolutionary Mattathias was preparing to die, he made a speech to his sons, exhorting them to zeal for the law, and invoking the zealous heroes of old. The speech ends as follows:
Judas Maccabaeus has been a mighty warrior from his youth; he shall command the army for you and fight the battle against the peoples. You shall rally around you all who observe the law, and avenge the wrong done to your people. Pay back the Gentiles in full, and obey the commands of the law.
With that, Mattathias died. The sequel, as we have seen, is that Judas took command, led the revolt, fought the battle, defeated the pagan army, cleansed and restored the Temple, refortified Jerusalem--and established a royal dynasty that lasted for a hundred years. Through Herod's marriage to Mariamne, it was still in existence in Jesus' own day.
Mattathias instructed his sons to give back to the pagans an equal repayment: do to them as they have done to us. The saying is unambiguously revolutionary. The second clause put this in its wider context: obey the command of the law. ... The Maccabaean saying had a double thrust: your duty towards the pagans is to fight them, and your duty to our god is to keep his commandments.
I propose that Jesus' cryptic saying should be understood as a coded and subversive echo of Mattathias' last words. His Temple-action created a context within which his saying would have meant: Pay Caesar back what he is owed! Render to Caesar what he deserves! The words Jesus said would, prima facie, have been heard as revolutionary.
When, however, the words are set in context, they acquire a second layer of meaning. Jesus was not in a classroom giving a lecture, or for that matter on a battlefield urging on the troops. He was facing a questioner with a Roman coin in his hand. Suddenly, a counterpoint appears beneath the coded revolutionary meaning; faced with the coin, and with the implicit question of revolution, Jesus says, in effect 'Well then, you'd better pay Caesar back as he deserves!' Had he told them to revolt? Had he told them to pay the tax? He had done neither. He had done both.