The Federal Reserve is a "two-headed" beast—a public federal agency in D.C. The Board of Governors, overseeing 12 private-ish regional banks owned by commercial member banks. The Board is a federal government department with a twist. They don’t get paid from the treasury, they make their own money from the U.S. bonds they hold. The 12 regional banks are privately held by large U.S. commercial banks which had to buy-in.

While the Fed doesn't decide to spend money (only Congress can do that), it does make it much easier for the government to borrow.

When the government spends more than it has, the Treasury issues bonds (debt). If there aren't enough private buyers, the Fed can step in and buy those bonds.

When the Fed buys those bonds, it doesn't use "saved" money ...