Poorer people are more obese is only true in richer countries or in countries where the poor doesn’t have to starve. As for why poor people are more obese in well-off countries. Again, it’s the path of least resistance. Poor people are generally less educated and less likely to think long term. It's easier to eat junk food than to put in the time and effort to make a well balance meal.
Btw, I'm not blaming the poor, being poor is hard. I grew up under the poverty line. Overcoming poverty is a huge task and more should be done.
Availability of cheap crap is much higher in more developed/high GDP countries I suppose.
In extremis, if you only eat what you can grow, rear, and trade for, then you're not eating breakfast ~cereal~ sugar, ready meals, crisps, etc.
It's a bit tangential, but this is similar to how such a lifestyle isn't even cheaper - it's not the only shopping those people can afford - most of my meals probably cost less, because I take an interest in cooking and eating (i.e. I'm buying vegetables not ready meals) but that doesn't mean there's always (or even that often really, especially amortised over all uses) an out-of-reach expensive ingredient.
Go to the dollar store and look at the food there. Everything has sugar in it.(It's also not a great deal due to portion size, but if you don't have much money it's an easy way to get variety). The cheaper the food, the more likely there are cheap sugars meant to made it more palatable
Or look at restaurant/fast food-- the healthier options are the 'premium' options. But if you've got a few buck and want to feed a whole family, a bunch of burgers is still reasonably priced for someone low income
Edit: still talking about the USA here.