Its all about what food goes in, compared to what is burned off. But there are many contributing factors that determin where the balance lies:
Low fat/high sugar 'healthy food'
High fat, easy to cook/collect from the take-away food
Portion size
Snacking
More sedentary life style
it isn't difficult to tip the balance in the direction that results in weight gain, by just not being conscious of exactly how much/what you are eating and doing, and that takes effort and self awareness.
I have a low income. I work from home part time and study part time. I have 3 kids, all of us are well within our healthy BMI's. My weekly food budget for the 4 of us is around £50. I cook from scratch all of the time with the exception of a very occasional chippy supper. The majority of the meals I cook are on the table within 30mins, or quickly prepared but slow cooked in the oven type dishes, not as quick and easy as chuck in the oven food, not quite as cheap as the cheapest Iceland specials but cheaper than ready meals and a damn site healthier, and easier to bulk up to keep tummies full.
I have a small veg garden and fruit trees and bushes that supplement what I have to buy, which helps keep the food bill down during the summer and autumn mainly, although it would do much better if I looked after it more! I always use full fat milk and proper butter.
We are all active (I am probably the least physically active of us all) but none of us are exercise mad, and i limit screen time for the younger two (4&7), though not for the eldest who is 16. If I have to put things back out of my on-line shopping basket when I spend too much -sadly a common occurance- the crisps, juice, biscuits and snacks are the first to go (much to DS's disgust) so often if we snack, it is on fruit. I am also conscious of portion sizes, and we only have pudding (that isn't fruit or a yogurt) maybe once a week, and meat 3-4 times a week. I do like baking, but as I have to fit it in around everything else there is only an intermittent supply of yummy cakes.
It totally is possible to eat well on a limited budget, however, to do so you have to cut out a lot of the luxuries and it is more hassle. The benefits I have noticed in my kids are not just that don't have weight problems, but also they rarely get ill, and if they are ill they all shake it off quickly, also there is only 1 tiny filling between the lot of them (DS, earlier this year after his braces were removed).
Man, that's a long post! Sorry 