When people save more money, the interest rate will tend to go down. The reason for this can be seen in supply and demand analysis. When people save, the supply of money available to be loaned goes up. As you know, when supply goes up, equilibrium price goes down, all other things being equal.
As you also probably know, the price of money is the interest rate. If the supply of money goes up, the price of money should go down. Since the interest rate is the price of money, the interest rate should go down when savings go up.
Relationship between rates of saving and interest can be understood in terms of relationship between supply, demand, and price of credit. The savings of individuals by way of deposits in banks and other forms of deposits constitute the supply of money on credit. The money thus available through savings is then loaned to borrowers. Greater is the willingness and ability of individuals to save, greater is the total amount saved and available for lending to borrowers. And when the supply of funds available with banks for lending is higher they need to lower the interest rates charged to the borrowers to induce them to borrow the increased funds available. At the same time when banks find that individuals are ready to deposit more funds than they can loan out at existing interest rates paid to depositors, they reduce the amount they are willing to pay as interest to the borrowers. In this way increased saving puts downward pressure on rates of interest paid by banks to depositors, as well as rates charged by them from borrowers.
However, it is important to note that the discussion above presents only one side of the supply and demand behaviour. The demand for credit for borrowers depends not only on interest rate but also on the general level of economic activity. Thus when economy is booming, both the demand for credit and interest rates may move upward simultaneously. Put another way, level of savings is one of the determinants of interest levels. The demand for credit because of general economic condition is another important determinant of interest rates.