This is it. You trigger a pull back in the economy, if you get lucky you don't trigger a recession. But, if you need to trigger a recession you do it rather than out of control inflation.

How exactly do rising interest rates reduce demand for food?

How does it reduce demand for gas?

This is what I mean.

Per other commenters less money in the economy reduces aggregate demand. Per previous inflationary periods where interest rates were raised basically by putting people out of work.