In theory, stimulus checks are intended to increase the amount of capital in the economy. By giving back tax dollars in the form of a stimulus check, governments hope that consumers will make purchases that will in turn provide business the capital they need to keep functioning. Even if the check is put in the bank, the logic goes, then the bank will have more deposited capital and will be able to lend more freely to businesses and individuals.
However, there are some problems with this theory. Banks may need to deleverage and will choose to hold on to deposited capital rather than loosen lending, as they have during credit crises. Similarly, companies may choose to hold onto profits to form a war chest in anticipation of an economic downturn rather than increase production and hire more workers. And lastly, the people getting the checks may also choose to use them against debt or hold them in guaranteed government securities, stimulating nothing, but improving their personal situation. In short, all parties will look to their own self-interests over that of the economy.
That doesn't mean that the principle of stimulus checks is completely without value. If, instead of a temporary measure in a single check, permanent tax cuts at corporate and individual levels allowed both parties to keep more of their earned income, there is a good chance that the flow of capital would eventually increase. The long-term cut would mean that all the players in the economy could recapitalize, that is eliminate debts and increase savings, on ongoing basis. A stimulus check tries to reduce this long-term treatment down to a quick shot-in-the-arm and consequently has the potential to fail from a long run perspective.
For more, see How do government-issued stimulus checks affect the economy? and Do Tax Cuts Stimulate The Economy?