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Read up on Paul Volcker and the federal reserve board strangling inflation with interest rates.

Volcker was a Carter appointee. It was the tough medicine needed at the time, and it cost Carter his job for doing the right thing.

"Break the back of inflation" was the phrase attributed to Volcker.

It won't work this time because of our 30 trillion debt. We can't afford higher interest rates.

Easy. Increasing interest rates does the following:
Increases the relative value of the USD relative to other currencies, making imports cheaper.
Curbs asset prices, especially anything highly leveraged like real estate, and possibly stocks (assuming buying stocks on margin is still a thing).
Reduces the value of existing bonds, but makes new issued bonds at the higher interest rates a more attractive investment, since I'd you can get a guaranteed 4% or more rate, lots of investors are going to avoid the headaches of real estate, and the risks of stocks (especially retirees).
At higher interest rates, less borrowing occurs.
Lending is highly inflationary because it artificially increases the money supply.

Of course, if the fed stopped buying mortgage backed securities, mortgage rates would normalize and housing prices would plummet. They could literally eliminate housing inflation overnight.

Booger says

Easy. Increasing interest rates does the following:
Increases the relative value of the USD relative to other currencies, making imports cheaper.
Curbs asset prices, especially anything highly leveraged like real estate, and possibly stocks (assuming buying stocks on margin is still a thing).
Reduces the value of existing bonds, but makes new issued bonds at the higher interest rates a more attractive investment, since I'd you can get a guaranteed 4% or more rate, lots of investors are going to avoid the headaches of real estate, and the risks of stocks (especially retirees).
At higher interest rates, less borrowing occurs.
Lending is highly inflationary because it artificially increases the money supply.

And chokes off credit. Businesses large and small cannot afford the payments for the interest rates for their expansion or even to fund day to day operations. Credit market "seizes up", as they say. Same for households and local government.

They don't. Inflation is caused by only one thing, printing money backed by nothing. Inflation will never go down as long as we have a central bank.

Now the above is in regards to attempts to manipulate interest rates. In terms of real interest rates as determined by the free market, they are a reflection of risk. When risk becomes too great, people lose confidence in a currencies ability to hold it's value, and thus repay debt. For a recent example see the overnight repo markets, especially right before the scamdemic. But for a real explanation, see Griffin.

I always looked at interest rates like:

The act of borrowing steals spending money / consumption from the future so that it can be spent today. As such, it behaves somewhat like printing.

Lower interest rates are more attractive to borrowers (obviously). Lowering rates will thus cause more borrowing. Raising will cause less.

So, lowering / raising rates has an effect kinda similar to printing in that is alters how much money enters the economy right now (by stealing from the future), and they adjust the amount "printed per month" via rates.

Secondarily, low rates also allows repayment of existing debt to be more easily deferred. So, low rates not only stimulate the addition of new money month, but it also reduces the amount of money that gets paid back per month. And a penny saved is a penny earned.

And, taking money out of the economy combats inflation by making people spend less as a way to reduce demand, because they borrow less.

it shrinks money supply by making leverage more expensive. less leverage = less money supply = inflation slowing.