The Federal Reserve does not print money--the U.S. Bureau of Engraving and Printing along with the U.S. Mint are responsible for that. The Federal Reserve, however, is responsible for putting these notes into circulation. This can be accomplished through open-market operations, which is basically the Fed buying and selling government bonds to the public. When it buys bonds, it puts more currency into the economy, thereby increasing the money supply. When it sells bonds, it takes currency away from circulation, and, consequently, decreases money supply.
Money supply manipulation is an important mission of the Federal Reserve, which acts as a central bank for the United States. You're right in indirectly pointing out that an increase in money supply leads to an increase in price levels (inflation), a decrease in the value of money, and a decrease in the interest rates (interest rates must fall to induce people to hold the additional money). When people have more money to spend, in the short run, they can use it to purchase more goods, thereby increasing the aggregate demand and helping avert a recession.