Higher interest rates doesn't reduce these prices directly. Higher interest rates tend to reduce the overall inflation rate through reducing aggregate demand in the economy (slower rate of economic growth). Therefore, higher interest rates reduce these prices indirectly (and there may be significant time lag.)
- If interest rates are increased, then consumers will face an increase in the cost of borrowing (mortgage payments will be higher) therefore, they will have less disposable income and so will reduce their spending (e.g. decide not to buy new TV because they need to spend more on mortgage payments)
- Also higher interest rates may make it more attractive to save money in the bank rather than spend.
- In the UK, higher interest rates are also likely to depress house prices. Because mortgages are more expensive, people may decide to sell and rent instead. Falling house prices leads to a decline in wealth and therefore discourages people from spending.
- Finally, higher interest rates are likely to lead to an appreciation in the value of the pound sterling. (It's more attractive to save money in UK banks because interest rates are relatively higher). Therefore, this increase in demand for sterling leads to an appreciation in the currency. When the currency increases in value, exports are more expensive leading to lower demand for UK exports.
- Depress wages (leading to lower wage growth and lower inflation).
- Encourage firms to discount unsold goods - putting downward pressure on prices.
This is why it is difficult for the Bank of England when there is an increase in cost push inflation (higher food / oil prices). They would like to have a lower inflation rate, but to reduce inflation will cause other (bigger) problems - i.e. higher unemployment and lower growth.
It was suggested that reducing the UK inflation rate after 2008, would have caused a prolonged depression (why we tolerate higher inflation). Therefore, I would say the Bank of England did the correct thing to tolerate higher inflation rather than cause higher unemployment.
Also, it is still possible that some prices may continue to rise, despite a fall in growth and lower overall inflation. The price of food and raw materials is determined partly by overall macro factors, but also individual supply and demand. If we run out of oil, the price of oil may continue to rise, no matter how high interest rates are.