Think about it like this. A lot of things are financed via credit/debt. For example, you buy a house, using a mortgage, which has an interest rate.
Some people might be able to finance a $500k house, at 0% interest rate for 30 years. However, with certainty there are fewer people who are able to finance a $500k house, at 10% interest rate for 30 years, as this costs much more. This means, as interest rates rise, fewer people will be able to secure and offer $500k for a house. As fewer people are offering $500k for a house, sellers will struggle more to sell a house for $500k, and eventually, there even comes a point where prices are forced to drop. As for how high interest rates need to get and how long they are in place to cause a decrease in prices, is mostly a guessing game, but eventually it will cause an effect.
Now, that said, initially people might try to raise their prices to offset the increase in financing costs, (and therefore adding to inflation like the parent post thought), but like I said, eventually there comes a point where they just won't find anyone who will buy their higher prices, because no one can afford those higher prices, and therefore prices must come down if transactions are to continue happening at all, and some transactions will surely continue to happen, as some money is better than no money, when you got a liability to pay.
And this also applies to food prices, along with most everything, in very similar but indirectly complex ways, mainly because most transactions in the economy happen via credit, then with actual money/cash. Let alone the fact that all money originates via central bank loans.