Jesus was being asked about the temple tax. The temple tax was a tax each Jew had to pay when they came to the temple. All Jews were required to come to the temple multiple times each year... so this was a direct tax on their meeting God's Commandments. Among the Jews of the time, this was a very contentious subject.
The Jews expected the Messiah to come, throw off the yoke of the Romans and reestablish the throne as a descendant of David's line. Jesus, on the other hand came as a spiritual, not political savior. As he latere stated to Pilate "My Kingdom is no part of this world".
So the tax statement by Jesus was defining a separation between 'church' and 'state'. Specifically, that his salvation, his kingdom and his followers were not there to overthrow the Romans, or protest unfair taxes. They were there to do God's Work. Thus, if Ceaser wanted the tax, they should pay the tax, so that they could focus on the spiritual, rather then being bogged down in the political.
A strict interpretation of Jesus gospel can build a strong argument that Christians should be politically neutral, focused instead on their relationship with God and their salvation through God's Kingdom.
Of course, thats assuming that the poor guy actually lived and said/did the stuff written about him.