One of the danger signs of an economy in trouble is when the personal debt rate climbs much faster than the savings or investment rate. Savings and investment rates are measures of consumer confidence and economic stability.
Investment rates also reflect the health of a nation's business environment, i.e. how easy it is to start a business, to obtain loans, to offer public stock and expect a favorable climate to start a business.
Savings rates also reflect whether or not a society is living within its means, which is an accurate predictor of long term economic viability.
In the long term, economic growth can only happen if there is investment in an economy. The investment allows the economy's production possibilities curve to move outward. This happens because firms can, for example, buy new machinery or develop new technologies when they have money to invest. These new things allow them to be more productive.
Savings are a very good source of money to be used by companies to do this kind of investment. If the people of a country can save sufficiently, the country's firms will not be dependent on foreign sources of investment.
So the relationship is that saving provides a source of money to invest. Investment allows for firms to buy new capital goods. This leads to economic growth.