With a higher interest rate, you may end up paying more interest over the life of the loan. To combat inflation, banks can set higher reserve requirements, there is a tight money supply, or there is a greater demand for credit. In an economy with high interest rates, people resort to saving money because they receive more than the savings rate. The stock market suffers, as investors prefer to take advantage of the higher savings rate than invest in the stock market with lower returns.
Companies also have limited access to equity finance through debt, leading to an economic contraction. The Federal Reserve's most important goal in raising interest rates is to slow down economic activity, but not too much. When rates rise, which means it's more expensive to borrow money, consumers react by abstaining from making large purchases and reducing their spending. The idea is that, in the current environment of high inflation, this decline in consumer demand could help bring prices back to normal.