If you want a simple understanding of this, it is not going to happen. The forces affecting the economy are so many and so varied, that it is an art as much as a science. Some effects happen in a few weeks, some in a few months, some over a year, and some decades or generations. And all those are pushing and pulling in different directions and on different sectors of the economy.
Trying to answer directly and simply would necessarily omit a lot of the nuances and subtleties, and feel really unsatisfying. And even the experts disagree on how things might turn out if you raise prime interest rate a little more or a little less or not at all.
Raising the prime rate (which is what the Fed does) cascades through the banking lending system and effects certain markets pretty fast (like housing since mortgage rates follow the prime rate -- sometimes quick, sometimes laggy, and sometimes even LEADING the Fed's actions). That affects certain behaviors with home buyers, which indirectly and later will lead to rents changing, and it can affect other big purchases like cars (where you need loans). The prime rate also bumps credit card interest rates, so people will want to (but may not be able to) spend less on their credit cards, so maybe they wait for other middling purchases like appliances or furniture -- slowing inflation on that sector). BUT, inflation makes people want to buy SOONER because if they wait, their $800 they have in the bank can buy the dishwasher today, but the same one might cost $900 next year.
And all the while, energy prices (which impact inflation and every other sector as well) are affected by other influences throughout the world like hurricanes, wars in Europe, OPEC actions and reactions, bringing refining capacity up or not... Energy prices affect how much people have left over after their gasoline and electricity bills, which are NOT tied as closely to the prime rate.
And of course there are seasonal effects, supply chain snafus still finding their way back from the pandemic (and still being effected by China's "No COVID Policy" which shuts down various economic centers in a chaotic fashion), and literally, the price of [tea? wheat? rice?] in China.
But overall, raising the prime rate can slow things down, eventually enough to get inflation under control (hopefully) without causing too painful of a depression. Because, in the opinion of most economists, runaway inflation and stagflation would be even worse than a depression.
I'm no economist but I've spent a few years reading Business Week, Forbes and the WSJ, and listen to Marketplace every afternoon on the way home from work and it lets me know how much I don't know, and how much of it is "best guess" work by the people actually making the decisions.
