When the inflation rate increases, this tends to push up interest rates because of supply and demand: If the interest rate is less than the inflation rate, then putting your money in the bank means that you are losing value every day that it is there. So there's an incentive to withdraw your money and spend it now. If, say, I'm planning to buy a car, and my savings are declining in real value, then if I buy a car today I can get a better car than if I wait until tomorrow. When interest rates are high compared to inflation, the reverse is true. My savings are increasing in value, so the longer I leave my money in the bank the more it's worth. If I wait until tomorrow to buy a car I can get a better car than I would be able to buy today. Also, people find alternative places to keep their savings. If a savings account will result in me losing value every day my money is there, then maybe I'll put the money in the stock market or buy gold or whatever.
So for the banks to continue to get enough money to make loans, they have to increase the interest rates they pay to lure customers back to the bank.
There is no reason per se for rising interest rates to consumers to directly cause an increase in the inflation rate. Inflation is caused by the money supply growing faster than the amount of goods and services produced. Interest rates are a cost. If interest rates go up, people will borrow less money and spend it on other things, but that has no direct effect on the total money supply.
Except ... you may note I put a bunch of qualifiers in that paragraph. In the United States, the Federal Reserve loans money to banks. It creates this money out of thin air. So when the interest that the Federal Reserve charges to the banks is low, the banks will borrow more from the Feds. As this money is created on the spot, this adds to the money supply, and thus contributes to inflation.
So if interest rates to consumers are low, this encourages people to borrow more money from the banks, which encourages the banks to borrow more from the Feds, which increases the money supply, which increases inflation.
I don't know much about how it works in other countries, but I think it's similar in most nations.