What’s the paradox that says increasing the price of bread makes the poor eat more of it?
It’s the sort of paradox that is “something that looks odd” rather than the sort that is “another word for contradiction.”
The idea is that bread is the example of something really cheap. Something that everyone can afford to eat. Something for which there aren’t substitutes at the same price point.
I’m going to make up numbers here, in a made-up currency called Bibbles, because I don’t want people saying “but bread doesn’t cost XXX”.
Bob’s current situation
Let’s say that:
- Bob has 9 Bibbles to spend on food each week.
- A loaf of bread costs 2 Bibbles
- A meat pie costs 5 Bibbles.
- If he has the money, Bob prefers pies over bread. 
- Bob needs to eat three “items” each week to actually survive.
Bob needs to eat enough, so buys 2 loaves of bread and 1 meat pie each week.
Now the price increase
Now bread goes up to 3 Bibbles.
Bob can no longer afford 2 loaves of bread and a meat pie. The only thing Bob can now afford is 3 loaves of bread.
So, because bread gets more expensive, poor Bob buys more of it.
Back in the real world
In many things in 21st century economies, we have alternatives. If whitecurrants double in price, more people will eat redcurrants instead… so the growers of whitecurrants can’t make out like bandits and increase the price every week…
… but in many situations, particularly for those literally struggling to earn enough to eat, there may well be a selection of goods that are the “cheapest” where there aren’t obvious substitutions.
 If this answer is read by any real economists, you can admire the way I skate over decreasing marginal utility there :-)