The Capitol always want to be in charge and superior to the Districts. A way of doing this is taking away their food. This means that the Districts are relying on the Capitol for food, meaning that the Capitol has an advantage over them. By not allowing them to hunt, the Capitol can make sure that the only food source comes from them.
In short...
A (the Capitol) wants B (the Districts) to reamin poor so that A can remain in charge a.k.a A wants to be greater than B.
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(SPOILER WARNING)
If you have read the new book (The Ballad of Songbirds and Snakes) then you obviously know the tragic ending --(again, if you are in the middle of reading it or have not purchased it yet then please consider not reading this)--of Lucy Gray in the middle of the woods. That might be the reason the woods are restricted to others. That someone identifies the bones (successfully) and finds out about the murder. You never know if some of them knew personally the Covey. Of course, I may be wrong on this subject because it isn't confirmed yet but still a pretty good theory.
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They don't want people escaping
a person who engages in an illegal business dealing
It's a place where they can go and trade their items for coins
The Food games The death games the fighting games Or the appetite games
The hunger games