The purpose of automatic stabilizers is to increase disposable income when an economy is in recession and to decrease it (to some extent) when an economy is in a period of expansion. Automatic stabilizers are not able to completely smooth out the business cycle, but they are meant to help flatten the cycles out -- to prevent recessions from being too deep and expansions from going too fast and causing inflation.
For example, unemployment insurance is an automatic stabilizer that kicks in during a recession. When people lose their jobs, these benefits increase their disposable income. The unemployed are then able to keep spending to some extent so that aggregate demand will not drop too much.
During an expansion, a progressive income tax acts as something of an automatic stabilizer. As incomes rise, the income tax takes some of that money away, decreasing disposable income and helping to prevent excessive inflation.