How higher interest rates bring down inflation.

 

The above graph shows a sharp drop in mortgage applications (to buy houses).  This is how the Federal Reserve is trying to reduce inflation:  by raising interest rates, they reduce demand for capital spending, e.g., equipment, cars, and houses, where you typically borrow to make the purchase. 

As demand for goods and services falls, their prices start falling as well.  From CalculatedRisk.com

Recently we've seen used car prices down over 10% and framing lumber prices down 33% 

Currently inflation is running close to 9%, and many expect the Fed to keep raising interest rates until the rate of annual price increases drops to around 2%.