The basic assumption is that the economy starts off with unused resources, for example some workers are unemployed. By increasing demand in the economy it is then possible to boost production. If the economy was already at full employment, any attempt to boost demand would only lead to inflation. Note also that even if, say, some workers are out of work, it may be difficult to employ them directly due to bottlenecks in other parts of the economy.
As an example, consider the government increasing its expenditure on roads by $1 million, without a corresponding increase in taxation. This sum would go to the road builders, who would distribute the money as wages and profits. The households and firms receiving these incomes will save part of the money and spend the rest. These expenditures in turn will generate more wages and profits, and so on with the money circulating around the economy.
The eventual amount by which output expands is governed by the marginal propensity to save, which is the proportion of extra income that is saved rather than consumed. If the marginal propensity to save is large, less money is returned into the economy with each circulation so the multiplier effect is smaller.
The value of the multiplier in a closed economy with no taxes is given by 1/s, where s is the marginal propensity to save. Taxes and imports tend to reduce the value of the multiplier ("leakage").