A built-in stabilizer works to slow the rise or fall in aggregate demand (AD) by giving more money to consumers or by taking money away from them. As the people have more or less ability to buy, AD is affected. If the government gives money to people in a recession, the people have more money and the decline in AD is likely to slow or stop. If the government takes from people in an overheated economy, they have less money to spend and the rise in AD is likely to slow or stop.
A good example of this is the effect of unemployment insurance benefits in the wake of the 2008 economic crash. When the crash happened, levels of unemployment jumped drastically. This reduced AD because the unemployed people had less money to spend. But unemployment benefits help to mitigate this decline in AD. People who became unemployed could get unemployment benefits that allowed them to keep spending to at least some degree. This meant that AD would not decline as sharply as it might otherwise have.
The same dynamic works in reverse with progressive taxes and an overheated economy. If people start to make much more money, AD will rise rapidly. But if a good deal of this money is taken away by progressive income taxes, the rise will be mitigated. In these ways, built-in stabilizers can help to reduce the swings in AD that come with the business cycle.