There are generally said to have been two motivations for the voyages of Zheng He. Neither of them, in contrast to European voyages of exploration, had to do with wealth or actual practical power over any other country.
One motivation for the voyages has to do with internal Chinese politics. Zheng He was working for the Emperor Yung-lo who had taken the throne in a coup. The previous emperor had not been captured but had, instead, run away. It was said that he might have gone to Southeast Asia. It is said that one reason for the voyages was to look for him.
However, this was clearly not the reason for most of the voyages. Zheng He did not spend much time looking for the ex-emperor, and the voyages to Africa were clearly not meant to find him. Instead, these voyages seemed to be meant to show off. They were meant to show the world the power of China. The emperor hoped to get people in other countries to acknowledge his importance and that of his empire. However, this would not mean that they would be ruled by China in fact.
Thus, the voyages of Zheng He were largely about securing prestige for China.