The concept of "velocity" is what many fail to understand. Consider this scenario:
The government hands me a check for $10. I use the ten dollars to have a pair of shoes repaired, the cobbler takes that $10 and pays the neighbor to shovel his sidewalk, and the kid takes the $10 and pays Blockbuster for two video game rentals. The total boost to the GDP is $30, assuming all of that income is reported--$10 for the cobbler, $10 for the kid, and $10 for Blockbuster.
But if I take the $10 and shove it under my mattress, the net boost to the GDP is $0.
This is why it is important to stimulate the economy by getting money into the hands of people who are most likely to spend it. Studies show that would be poor people. It doesn't have to be hand-outs. It could be by figuring out how to get more of the unemployed poor people working again.