The answer to this question can be found in the first chapter of this gripping dystopian novel. As part of the general background that Katniss gives her about District Twelve and the kind of lives of hardship and poverty that she and so many others lead, she also tells us about the creation of the Hunger Games and how they are designed to maintain control over the Districts and ensure that no other rebellion occurs. The ever-present memory of the complete destruction of District Thirteen thus acts as a powerful reminder of the price to pay for any who dissent. Note what Katniss tells us about the benefits that Districts can gain if one of their children wins the games:

To make it humiliating as well as torturous, the Capitol requires us to treat the Hunger Games as a festivity, a sporting event pitting every district against the others. The last tribute alive receives a life of ease back home, and their district will be showered with prizes, largely consisting of food. All year, the Capitol will show the winning district gifts of grain and oil and even delicacies like sugar while the rest of us battle starvation.

Thus we can see that having a representative from your District winning the Hunger Games yields immense rewards to the District, and they gain extra food rations and even luxury items such as sugar as a result.