I know automatic stabilisers can be a little confusing, so I'll try to explain it carefully.
During a boom, economic growth is very high, probably above 4%, full employment is being achieved, and inflation is likely well above the target. During this period, those in the workforce generally earn greater income. As a result, they will have to pay more tax and may shift into higher tax brackets. This means that disposable income should rise slower. As a result, aggregate demand will rise slower and the economy overall will slow down, meaning that economic growth contracts.
In a period of negative or very slow economic growth, such as a contraction, depression, recession or trough, the population will generally be in the lower tax brackets, so the government collects less revenue. This essentially means that there should be greater disposable income, as opposed to collecting high rates of income tax. As the population has more money to spend, aggregate demand should strengthen, which will speed up economic growth.
I feel like that explanation is a bti confusing so, simply, during a boom, people pay more tax, so they don't have as much money to spend, so aggregate demand will decrease. During a downturn, people pay less tax, so they have more money to spend, which should increase aggregate demand.