In general, as interest rates are reduced, more people are able to borrow more money. The result is that consumers have more money to spend, causing the economy to grow and inflation to increase. The opposite holds true for rising interest rates. As interest rates are increased, consumers tend to save as returns from savings are higher. With less disposable income being spent as a result of the increase in the interest rate, the economy slows and inflation decreases.
A fall in the inflation rate could cause various benefits for the economy:
Goods of that country becoming more internationally competitive increasing exports and growth
Increase rates of return for savers
Improved confidence, encouraging firms to invest and boost long-term economic growth.
Increased disposable incomes (if nominal wage growth is constant)
However, if a fall in the inflation rate is due to depressed demand, it could create deflationary pressures which make it hard to boost economic growth. It is worth bearing in mind governments usually target an inflation rate of 2%. If inflation falls from 10% to 2%, this will tend to have economic benefits. If inflation falls from 3% to 0%, this could indicate a depressed economy.