Given the rampant hunger by many in Katniss' District 12, why aren't more citizens breaking the law and venturing into the restricted areas in search of food?
It's a combination of two factors:
"Restricted" areas have dangerous predatory wildlife. To venture there even remotely consistently for a meaningful length of time, you absolutely must be armed with a real stand-off weapon, or you are lunch for something toothier.
And of course, the Capitol would never allow people in the districts to be armed with real stand-off weapons (and not everyone is able to make their own). Peace-keepers sometimes tolerate someone who is able to make one for themselves and hide it in the forest - but few people are good enough to make their own weapons and making weapons for others is a Big Bad Thing Not To Do - that's how rebellions become viable.
Both of those points are explicitly explained in the very start of "Hunger Games", by Katniss (emphasis mine, the most critical sentence is italics+bold):
Electrified or not, the fence has been successful at keeping the flesh-eaters out of District 12. Inside the woods they roam freely, and there are added concerns like venomous snakes, rabid animals, and no real paths to follow. ...
Even though trespassing in the woods is illegal and poaching carries the severest of penalties, more people would risk it if they had weapons.
But most are not bold enough to venture out with just a knife. My bow is a rarity, crafted by my father along with a few others that I keep well hidden in the woods, carefully wrapped in waterproof covers. My father could have made good money selling them, but if the officials found out he would have been publicly executed for inciting a rebellion. Most of the Peacekeepers turn a blind eye to the few of us who hunt because they’re as hungry for fresh meat as anybody is. In fact, they’re among our best customers. But the idea that someone might be arming the Seam would never have been allowed.
In the fall, a few brave souls sneak into the woods to harvest apples. But always in sight of the Meadow. Always close enough to run back to the safety of District 12 if trouble arises.
The capital severely punishes people who head out to the restricted areas (e.g. death, or turning people into Avoxes). And the last known wide-spread revolt was District 13, which was destroyed as a result. So people instead opt to live a hard life within the capital's rules, mostly. In District 12, there is a thriving black market that helps the people maintain easier lives than what the capital allows. It's unknown if other districts have similar back-alley markets.
Thanks to the black market, only a few people have to use their skills to retrieve food from the restricted area. That's why so few do it, because it's not necessary for more people to take that risk.
There is a relatively narrow set of people who are eligible to be wandering in the restricted areas. These people must:
- Live in an area where the fences are not always on.
- Be skilled at hunting.
- Be willing to break the law.
- Have access to weapons, or be able to make them.
We only really know about the people from District 12, and of those, very few have the last 3. It is probably the lack of access to weapons that keeps most people from doing such, or the fear of getting caught.