High discount rate reduces the propensity for work (most poor don't work much, if at all), particularly work with a career path. Why work hard now just to have a middle class life in the future?
Similarly, a high discount rate cuts against a healthy lifestyle - why suffer through a run in the park and hunger right now, just for future health?
(Also explains correlation between poverty and: drugs, alchohol, unprotected sex, probably other pathologies I can't think of.)
High discount rate is the cause, everything else is an effect.
Of course, a high discount rate has nothing to do with the cheapness of carbs, fat and sugar relative to meat or veggies. Money is not the limiting factor. Willpower/preferences are.
No one is saying that poor people couldn't eat healthy if they wanted to more than they wanted to feel satisfied. They're saying that given two people with the same metabolic genes, exercise level, and discount rate, if one of these two people spends ten times as much on food as the other, they will be likely to be thinner. This is because being sated with more expensive food results in eating fewer calories and better nutritionally than being sated with less expensive food, and this point has everything to do with the cheapness of carbs, fat, and sugar (which taste good no matter how prepared, essentially) relative to well-prepared meat and veggies.
When people say that obesity goes with poverty, they're noting that virtually everyone eats until they are sated, and if you eat until you are sated with high-quality, expensive food, you will be far less likely to be overweight than if you eat until you are sated with low-quality, inexpensive food.