The Federal Reserve is the body that, among other things, oversees the monetary policy of the United States. Therefore, its role with regard to getting the country out of a recession is to set monetary policy that will be conducive to creating economic growth.
The Federal Reserve has to tools that it typically uses with regard to monetary policy. First, it can manipulate the interest rates that are charged by lenders in the US. The interest rate is, in effect, the price of borrowing money. When the Fed believes that the economy needs a boost, it can reduce interest rates. When it does this, borrowing increases because a drop in the price of doing something generally leads to an increase in that activity. Increased borrowing is good because it leads to more buying of consumer goods and capital goods.
A second thing that the Fed can do is to buy government securities. When the Fed does this, it is essentially printing money. It is using money that did not previously exist to buy securities from banks in the private sector. This puts more money into the economy. With more money in the economy, it is easier for consumers and businesses to borrow and more economic activity ensues.
Thus, the Fed’s role is to increase the money supply so as to encourage borrowing and spending. By doing these things, it can attempt to get the country out of a recession.