Definitely seems odd on its face, but if you dig into it, you'll see that it's SO much more complex than simple access. Other factors include, but are certainly not limited to:
Time to prepare: fresh foods generally take a lot longer, and a lot more effort to prepare than processed foods. When you've got each available parent working two jobs then this is a huge barrier.
Knowledge of preparation: especially in immigrant populations, not everybody has access to the ingredients they're used to.
Knowledge of nutrition: it's not like life comes with an instruction manual, and if you got little to no nutrition education, you probably don't realize the impact of those empty calories.
Fussy Kids: people who appeasing fussy kids (yeah yeah yeah... the hardos without kids will say you should just force them to eat it. Life isn't so straightforward.)
Waste: this is a huge factor. If you've got a tiny amount of cash and you need to spend it on things that will ensure you wont starve, something that will potentially go bad in a few days is a risky proposition, so you tend to stock up on heavily processed shelf stable items.
Price: Cheap is in the eye of the beholder. You can get a whole bag of chips for the same price as 2 apples. When you have a REALLY borderline income, your "cheap" is a lot different than someone else's.
There's plenty of other factors. Check out the research on it for more info. Really interesting stuff to dig into.