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There are several mechanisms:

1. choice: The idea is that you can put your money in the bank and get some interest or you can spend it. So when interest rates go up, people will chose to spend less and save more. Obviously rates have to be higher than inflation which is now 10%, for this strategy to work, but at high enough rates, they will choose to save rather then spend. Less spending, demand falls, so prices should drop.

2. money supply: For the non-financial sector of the economy, money is created when households borrow from banks, and money is destroyed when loans are repaid. Increasing interest rates reduces loan growth and thus the money supply. A smaller money supply should lead to lower prices.

3. business investment: higher interest rates means that the cost of capital to firms goes up. They must earn a higher margin in order to service whatever debt they have at the higher rates, and investors can choose to invest in the business or buy a bond, and so the business has to earn a return at least as high as the bond. So higher interest rates means that ventures which would have been profitable at a lower rate are no longer profitable. So less business investment at higher rates.