In what way can government spend the country into prosperity?
A country can't spend a country into prosperity but they can keep the recession from deepening. There are also ways of stimulating the economy such as infrastructure spending, which are very beneficial to the local economies: All of the $$$ are paid to local/state governments who are know the needs in their jurisdictions; The contracts are tendered locally and involve local workers and local material suppliers;
The public gets a permanent benefit for the dollars expended. I can't emphasize this last point too strongly.
The contractors, the suppliers and their respective workers all receive benefit from the monies they're paid for doing the work, and they all pay taxes on those monies so a portion of the federal government's expenditure on the contract will come back to the feds in the form of income and payroll taxes. But then those workers will take their net pay out into their communities to provide for their families, and the stores and service providers where they spend their $$ will also benefit, as will their workers. And some of those dollars will flow into state coffers as valued added taxes.
The government can also help young entrepreneurs launch new technologies (priming the pump, so to speak). I've seen the numbers on some of these programs and they're returning $1.50 in future revenues for every $1 they expend. I know of one young entrepreneur (under 25), who invented and patented a new technology which is now in use in big box stores nationwide. He received a $50,000 grant to move his technology into production once he had secured his patents, and now has over 200 employees in two manufacturing facilities.
The economy overall benefits enormously from the Defence Department: A large standing army reduces domestic unemployment, although in conservative parlance, all those soldiers and military personnel are "takers" because they receive free medical care, and other entitlements, and they pay no income tax. In addition, there is the entire military-industrial complex which provides the US with all of those wonderful and horribly expensive weapons they use, as well as the more mundane military necessities such as uniforms, office supplies, toilet paper, meals, computers, and on and on.
Additionally, there are the costs of maintaining the bases, both in the US and abroad. I can remember communities mounting huge campaigns to keep bases open when the Pentagon has suggested closing bases. They cite the number of local civilians the base employs and who would be unemployed, (electricians, maintenance staff, teachers, etc.), the losses to local businesses supplying the bases with goods and services, and the loss of business from military personnel living on the base. Military families buy food, clothing, send their kids to school, and again all of those value added taxes go to state and local governments.
The government gets pretty good bang for it's buck from the food stamps program too. All of that money gets spent locally, providing income for local grocery stores and their workers, and local farmers as well as other suppliers to the grocery stores. This affects truckers who deliver goods to the grocery stores as well. All of these people are getting some of that food stamp money and they're all paying taxes on that income so a portion goes back to the feds, and they're spending it to support their families and paying state taxes on their purchases and so on and so on.