The base rate is one of the very few parameters that is directly controllable. Which makes it a blunt instrument. When the only tool you have is a hammer, everything looks like a nail.
The Fed essentially controls the cost of money. Borrowed money can be used for many things. If money is cheap, it becomes easy to borrow larger amounts, and buy things for more money. Which has an inflationary effect.
For ordinary people buying a house is the big ticket item. Housing prices are highly dependant on the buyer’s income, and when the interest rates are low, people can borrow a lot more within that income and still be able to pay the interest. So more people have more money chasing a limited number of properties and driving up housing prices. Housing prices outstripping inflation is not a good thing, and eventually pushes many people out of the market. Further, continually increasing prices and cheap money makes real estate an attractive place for investment. Which further drives ordinary people out of the market. If you have already locked in a mortgage, rising interest rates hurt, but it isn’t you that is being targeted.
But this the only problem. Cheap money encourages all manner of investment behaviour. On the good side it encourages businesses to finance growth, which is good for the economy. On the bad side, cheap money encourages leveraged investments in speculative stocks and schemes. This can and does create bubbles. Any fool can make money in a rising market. An even bigger fool can borrow lots of money to invest in a rising market. (The biggest fool is the bank that lends the money to him.) A stock market roaring ahead really isn’t a sign of a healthy economy. It is often a sign of over eager speculation. A stock market (or any other market) bubble will eventually break, and that brings with it much worse economic pain. Increasing the cost of money, will, in theory, dampen down leveraged speculative investing. Sometimes it works.
As noted above - the problem is that the US economy - or indeed any modern economy - does not live in isolation. The Fed playing with the base rate does nothing to control the external factors driving the economy, or factors that are local but beyond its control. So inflation caused by spiking oil prices or large tariffs on goods won’t be fixed by raising rates. But inflationary behaviour caused by the availability of cheap money is. So choose your battles, and decide what you want to address.
The spectre of inflationary spirals as described by posters above remains in the minds of reserve banks worldwide. We haven’t seen one in a few decades, but the need to stamp one out early will figure large in the thoughts of the Fed. Recent history has seen the Fed act so more than once, and do so with good outcomes despite the significant short term pain. This rise is a drop in the ocean compared to those interventions. Whether such a tiny tweak as seen now will manage to turn the trend is another matter. I would view this more as the Fed putting a flag in the ground rather than a move that is itself expected to fix things. If indeed it is fixable at all.
Raising rates is a bad sign, not because raising rates is bad, but it is a sign that the Fed perceives that the economy is not operating well. (IMHO it has been obvious for a while that this is the case.)
One might note that very low interest rates are also a sign of a bad economy. Japan is the obvious example. Decades of economic stagnation, and the Japanese central bank has cut interest rates to essentially zero. And still nobody borrows to invest into building the economy. The once mighty juggernaught is moribund.