Basically the key is to stimulate inflation so that they get the chance to hike rates. By doing so many people choose to stop investing in stocks, funds and more on longer term saving accounts.
Essentially this heals the asset-liability structure of financial institutions. They now have liabilities of longer term so they can issue longer term loans. This eventually will stimulate real economy.
The fed didn't stimulate inflation by choice. The rates were at 0 for most of the decade and inflation was very low.
Increasing interest rates was a response to inflation -- partially created by QE and stimulus spending, supply chain disruptions, and other supply/demand shifts.
As for how interest rates work: they make borrowing more expensive. There's a bunch of second-order effects from that but borrowing cost and time value of money are the main thing to keep in mind.
EDIT: Fixed correlation sign.
Historically the Fed’s job was to “take away the punch bowl” and wring the bad debt and malinvestment out of the system before it became a systemic risk. Since 2008 the system has primarily focused on how to move the bad debt around to avoid default.