The economy runs like a circular flow of money and physical goods. What is produced must be purchased to sustain the current level of economic activities. In other words, what is not consumed (leakages) in the income stream must be plowed into the economy in the form of investment (injections). Now, realized (actual) leakages from the spending streams and realized (actual) injections into the spending streams are what actually happened. They must be equal by necessity. For example, goods actually supplied must be equal to goods actually demanded. What was not sold went into unsold inventory. In other words, the producers got stuck with the unsold goods. Because actual sales fell short of planned sales, the producers will adjust the planned sales downward and produce less the next period hoping that the actual sales will be equal to the planned sales.
In a simplified closed economy with no government, households plan to save a certain amount (planned leakage), but the actual saving (actual leakage) may be less than planned if producers do not plan to invest (planned injection) the same amount as the planned saving. When the planned injection is less than the planned leakage, households are disappointed in their actual saving and try to save even more next period. If the planned investment again does not match the planned saving, the output pie will shrink even more.
Thus collectively, the economy cannot save more unless it also invests more to soak up the higher saving. This is the paradox of thrift in a nutshell.
But government deficit spending (i.e., spending more than it taxes) can make up the shortfall of planned private investment, especially during economic recessions.
In an open economy, planned domestic leakages need not be equal to planned domestic injections to stabilize the economy. If planned domestic leakages exceed planned domestic injections, the excess domestic output could be exported to other countries. The resulting trade surplus could then be lent to countries whose total domestic spending (consumption, investment and government spending) exceeds their total domestic output. In turn, the incurrence of trade deficit allows some countries' planned domestic injections to exceed their planned domestic leakages. Thus, in an open economy, planned leakages need not be equal to planned injections in each country as long as they are equal globally through capital flows among them.
For example, as a result of foreign capital inflow, the US economy is actually larger than it can domestically sustain. But this is possible only by chalking up the US national debt.