Increasing interest rates

Why do they think raising interest rates will lower ‘inflation’ (which doesn’t include rising interest payments, mortgage costs and for education, student loans, child-care, local taxes, and skyrocketing drug prices)? Why do politicians fixated on deficits and interest rates think they will burden their children, but respond by cutting public services, jobs, government investments in maintaining vital infrastructure for our future in the name of ‘austerity’?