How does the Capitol use the Hunger Games to control the Districts?
There are several ways in which the Capitol uses the Hunger Games to control the people of the Districts.
As Zumba96 mentioned, the Games were instituted after the Districts rebelled against the Capitol seventy-four years prior to the events described in the novel. The Hunger Games is the way for the Capitol to showcase its power: it can take the children and execute them in the most brutal way—by making them kill one another—and the parents can do nothing but stand by and watch helplessly. To add insult to injury, the Districts are supposed to not only tolerate, but celebrate the Games, pretending that this is a joyous occasion. The Games are a reminder that if the Districts rebel again, there is no limit to the cruelty with which they will be punished.
Additionally, the Games control the lives of the citizens of Panem every day, not just during the Reaping (picking the participants) and its ensuing fight to death. The odds of being picked to be a Tribute (a participant) increase with the age of a child. A twelve year old gets one entry, a thirteen year old gets two entries, and so on. This is done to give the Reaping process an appearance of fairness—the younger the child, the harder for him/her to survive the Games, so the younger children have fewer chances of being picked.
But, there is a catch. People in the majority of Districts are poor and a lot of the families are starving. The children have an option of buying extra grain and oil for their families—at the price of getting an extra entry into the Games. Gale, who supports a single mother and younger siblings, admits that he had six entries when he was twelve. Now that he is sixteen, he has forty-two. So the citizens of the poor Districts are constantly faced with the decision of either increasing the odds of losing their children or dying of hunger. A victor, however, receives enough provisions to live comfortably for the rest of his or her life. Thus, the Capitol denies the people of the Districts ability to provide for themselves, and uses their desperation to manipulate them through Games. Hence, the title "Hunger Games."
In the novel The Hunger Games, the Capitol uses the games to control the people of the Districts.
The main reason the Capital can use the games to control the people of the Districts is fear. The people of the Districts are fearful that if they revolt against the Capitol, in protest of the Games, that the Capitol will destroy their respective districts as they had in the past with District Thirteen.
The Capitol uses District Thirteen to remind the other districts that they hold and maintain power given they can destroy any district at any time.
Another way the Capitol could maintain power over the districts is by withholding supplies from them. While this is not stated in the first novel, is is certainly inferred by the way the people are fearful of losing the supplies that they need to survive. As one gets further from the Capitol, it is apparent that the districts become poorer. The Capitol uses this fear to suppress any revolts which may come about as a result of protesting the games.
Every year a boy and a girl from each district is picked and chosen to participate in the Hunger Games to fight to the death which is broadcasted nationally. This happened after District 13 rebelled against the Capitol in history. The Capitol does this barbaric act to prove the consequences of a rebellion and how the other districts are powerless.