Lower interest rates mean businesses can borrow money cheaply to expand, add employees, conduct R&D for new products....etc
When the investors believe businesses are growing they purchase more shares of their stock.
This causes the stock prices to rise.
Another component to lower interest rates is people are more likely to take out home loans or refinance their existing home loan and used the savings to purchase items from companies which is the engine of the economy.
On the other hand when interest rates are high people are reluctant to borrow money and in fact there is more of an incentive to save money.
Right now people who save money would kill to get a 5% return on bank CDs! When a large segment of the populations elects to save money rather than spend it there is usually a recession soon afterwards. Sometimes people choose to save money when they hear about numerous layoffs because they fear losing their own jobs. Less money being spent in the economy makes stocks crash. Low interest rates spurs spending which stimulates the economy. Now is a great time to buy a home or new car! :)