Thanks for posting the link. I took a look at the table you are referring to. There isn't a simple pattern. Farmers intrinsically have better food security and few could be said to be in poverty. Economically, they are small business owners and have to be reasonably successful or they don't stay farmers for very long. But they only make up less than 2% of the US population. Even in a rural state such as Wyoming, most people are not farmers or ranchers.
Rather, compare it to this: https://en.wikipedia.org/wiki/Income_quintiles and this: http://www.fns.usda.gov/snap/income-rules-income-limits
You'll see that states that have lower median income (more precisely, more people in the first quartile) have more people on SNAP. The cause for all the variation is due to a thousand different reasons.
One of the reasons so many people qualify for SNAP is that you don't have to be extremely poor to qualify. A family of 4 can earn $30,000 a year and qualify. Being poor in the US is not like being poor in 90% of the world.
I've had to make ends meet here on that kind of income, and it's not easy. My wife and kids qualified for a state food program for a while (bread, beans, milk). But even though they got the food, and it helped our budget, we would have survived without. Others might not.
Yes, we have federal, state, local, community, and even church programs that help people. These are independent of welfare benefits. Much of the fight you are seeing is due to there being so many programs. Many disagreements as to efficiency and efficacy. What is the right delivery mechanism, how much funding for each, who is in control, etc, etc.
Like many American things, it's a mess, we enjoy arguing about it, and things manage to work.