Part of that demand is cash to buy and cost of the goods. If debt costs more then people have less to spend on other things. Like for instance higher credit card rates or student loans. That impacts purchasing power.
Inelastic goods may not be fixed by higher interest rates, especially with other supply chain/geopolitical/environmental impacts and companies debt requirements.
Elastic goods or investments, higher interest rates for loans for larger purchases or business purchases/investments/inventory certainly are impacted by higher rates when debt is involved.
In the end if the higher interest rates affect mostly business/consumer elastic goods then yes interest rates bring down price inflation and affects demand eventually. The flip is if the interest rate increases start to affect the supply like margins or business survivability, then any reduction in demand will just offset with a reduction in supply.