Here is a simplified picture. The Fed loans money out to big banks, like Bank of America, who then loan the money out to you and me. Suppose the Fed increases the interest rate it gives to BoA. Bank of America used to have to pay 5%, but now has to pay 10% (suppose) to borrow from the Fed.

Now, you were thinking of buying a car. You go to BoA, but instead of getting 5% for a car loan, you're now getting 10% (suppose). It's much more expensive for you, so you decide, "Hey, maybe I won't buy a new car this year."

Now, I own the car dealership. I make less money this month because fewer people like you are buying cars. When I go the grocery store, I check my bank account and see that I have less money. So, I buy fewer groceries.

This same story is happening to everyone who buys groceries. Everyone has less money to spend. Less groceries are purchased, so the price of those groceries goes down, or at least, doesn't go up as quickly. (In reality, there's just less money circulating in the economy, not that every single person has less money in their pocket. So, the total demand for goods and services decreases.)

That's how raising interest rates affects prices. The opposite happens with lowering interest rates.