Everything in an economy in interrelated with each other and in a broad sense the economies of the world are linked with each other as well. Countries export and import with each other and this is why if one country has a problem in its economy the other country will get affected as well. Just look at how the whole world felt the ripples of the effect of the collapse of the US housing market. This phenomenon is known as the ripple effect or the multiplier effect. Likewise, interest rates and inflation both are very strongly linked.
Lets first define inflation and interest rates just so that everyone is on the same page; inflation is defined as a general increase in the prices of commodities over a period of time. Interest rates are the percentage at which you borrow money, meaning if you borrow a set amount of money you will have to pay back more than your borrowed amount, this is due the value of money decreases over time.
The best way to understand the concept of the relationship between interest rates and inflation is with an example, so let's say interest rates in your economy have fallen, it gets cheaper to borrow from banks, getting credit cards, loans, and everything. You see people around you getting loans and using credit cards, and it compels you to think, why should not I? As a result, you get involved in bank borrowing as well, taking advantage of the interest rates, life seems great initially, you are able to pay your debts and monthly payments on time and you get used to it. However, after sometimes the case does not remain the same due to changes in demand and supply. You need to realize that time changes and as it passes, demand for everything will be so high that there would not be enough supply to meet that demand. For example everyone now has a car or a motorbike, and the demand for petrol has risen so much that supply becomes insufficient, and when this happens we see an increase in price because people are being willing to pay higher prices to get it, and this is when things start to go wrong. Now imagine every good and service starts to face this same problem, everything will become expensive and even if some commodities do not face this problem of increased demand they will have to increase prices because in general prices have risen and that's affecting their income as well. This is known as demand pull inflation.
Similarly when interest rates increase, borrowing becomes expensive and people save rather than spend because when they save, the same interest rate applies to their savings and saving looks a better option. This event results in a decrease in demand and when there is less demand in the market it leads to an excess of supply which force prices to decrease and inflation levels go down. And that's how interest rates and inflation are connected with each other.