Policy interest rates have wide reaching impacts across society, sometimes second and third order effects that can be hard to predict. Generally though, increasing interest rates encourages more saving instead of borrowing. Less borrowing means less aggregate spending. The overall effect is that people buy less stuff (lower demand). This lowers prices if the supply stays the same.
If it helps anyone at all, I recently published a new post on rising interest rates & what effects they might have: https://ericvanular.com/rising-interest-rates