Tax cuts do not stimulate the economy by "putting money in people's pockets" which they can then spend, as even some Republicans, including George Bush, mistakenly say. That's an old-fashioned Keynesian strategy, and, if it worked, the same result could be achieved by sending out increased welfare checks, which also puts money in people's pockets, which they can spend. But it doesn't work, because it doesn't do anything to change the basic incentives governing the economy, and because just borrowing money and then sending it out to people, in "tax rebate" checks or welfare checks, doesn't add anything to the economy on net.
Tax cuts stimulate the economy when they involve reductions in tax rates. The reduction in rates improves incentives for savings, investment, business creation and expansion, job creation, entrepreneurship, and work, by allowing people to keep a greater percentage of the reward produced by these activities. This improves the economy not just by the dollar amount of the tax cut. The improved incentives affect every economic decision and every dollar in the entire economy. The astoundingly successful Reagan tax cuts in the 1980s, as well as the astoundingly successful Kennedy tax cuts of the 1960s, were both based on reducing tax rates, and were successful for these reasons.