There are two different mechanisms by which a tax cut is supposed to be able to help revive an economy. First, there is the Keynesian, demand-side mechanism. Second, there is the supply-side mechanism.
According to Keynesians, a reduction in taxes will increase aggregate demand. This will happen because people will have more money to spend (because it was not taken in taxes). An increase in aggregate demand increases GDP, all other things being equal.
Supply-siders believe that tax cuts (especially for the rich and for businesses) increase supply. When taxes are lowered, people are encouraged to work and invest more because they get to keep more of their money. This increase in work and investment translates to an increase in aggregate supply.
Tax cuts help revive the economy simply in that the consumer will have more money in their pocket to purchase products. The more the consumer spends, the more products need to be produced, the better the economy.