QE puts money into the economy.
The Fed prints money and then uses the printed money to buy bonds in the secondary market. The fed receives the asset, the seller receives the printed money, which they then go onto spend. That's how printed money enters the economy and how QE expands the money supply. The Fed's reserves of assets expand simultaneously as the printed money enters the economy. They printed a lot during the pandemic, which is why we have inflation.
RE: Federal Reserve interest rates compared to inflation