Keeping the Roman Empire afloat was costly. Think about the buildings, the armies, all the layers of administrators, and even the costly banquets by the rulers. Occupied territories were taxed heavily to cover the costs of their occupation.
Coins were the form of currency. They carried the image of the emperor along with an assurance of his divinity.
Those who were plotting against Jesus tried to entrap him by asking if he thought it was legal to pay tribute to the emperor--that is, to use such a coin to support such an occupier.
Is his answer clear? Is he ending the discussion or is he starting one? After all, in Matthew's gospel, the Kingdom of God is not walled off in some other lifetime, it is a promise of what is happening here among us.
(with thanks to Preaching the Gospels without Blaming the Jews, Allen & Williamson)
Lectio Divina: Psalm 96:4