A cut in income tax rates would be one part of a classic Keynesian fiscal policy approach to combatting a recession.
Keynesians emphasize the importance of aggregate demand in macroeconomics. That means that, to them, an income tax cut is important because it will increase aggregate demand.
It will do this by increasing consumption. A tax cut will give people a higher disposable income. By definition, looking at the equation for the consumption function, an increase in disposable income makes for an increase in consumption.
This is because C = a + c Yd where
a is autonomous spending, c is mpc and Yd is discretionary income.
If Yd goes up and everything else is constant, C (consumption) must go up as well.
A cut in income tax will led to increase in net disposable income of people. And increase in disposable directly leads to to increase in private consumption. Therefore, a cut in income tax will directly lead to increase in consumption.
I think, I should explain a few additional points which need clarification. The increase in private consumption is explained by the principles of classical economics also. We do not need to make use of Keynesian economics to support the logic explained above.
Second, a cut in income tax will lead to increased disposable income, but it will also reduce the the amount of government revenue available for government spending. Thus tax cut by itself may not lead to increased economic activity in a country.