We hear in so many places that the current high levels of inflation are mainly due to supply side problems, like the war in Ukraine. Given that there is currently less supply then we are all inevitably going to be less well off in real terms. So why are central banks raising interest rates to "fight inflation"? The central banks can not shield the population from the inevitable drop in living standards. What do they think they can achieve?
It's a common misconception that central banks fight, or need to fight, current excess inflation. They don't, and they can't. It's already here and too late to do anything about it and real incomes are hit. However, central banks can do something about future inflation.
Indeed, there is a link between current and future levels of inflation. The longer high inflation lasts, the greater the chances high inflation persists into the future unless the central bank can successfully intervene. Current high inflation levels may translate into employees asking for wage increases, and higher input costs (wages and other input prices) may underpin future inflation, and so on.
So, central banks are targeting future medium-term inflation, not current inflation. Although it's not always explicitly stated, it's implied by how inflation and monetary policy work, the latter taking several quarters to develop its full impact. For central banks mandates, see, here for the Fed and here for the Bank of England.
A key element in these efforts to achieve medium-term inflation close to the target is to manage inflation expectations, based on which businesses set prices and wages are negotiated between employers and employees (among other things). By raising policy rates, central banks signal to the public and the markets their commitment to controlling future inflation, thereby already curbing inflation expectations if successful. Higher policy rates also act to cool down future economic activity directly by generally tightening financing conditions, which in turn reduces inflationary pressures on future prices, other things equal, helping to bring inflation back to target.
Note that central banks are aware that they are inflicting additional harm in the near term when they raise rates. However, they argue that the benefits of having low and stable inflation in the medium term as a result of these hikes outweigh the short-term pain.
Not sure about the source of your information, supply bottlenecks have not completely eased yet (E.g. China zero Covid policy). Also, inflation is not only/mainly (goods-)supply-side related : E.g. Money was made very cheap during the pandemic, labour markets being buoyant owing to fierce demand for labour after a brutal shutdown of the economy.
One of the mandates of a central bank is to avoid inflation spiralling (inflation feeds more inflation) . Again, this is a multi faceted-issue, slowing down the economy by increasing rates will soften demand which in turn is supposed to bring down inflation. Central banks walk on a tightrope between killing growth and protecting society from inflation (especially because the effect of hiking rates takes time to show symptoms in the economy), let alone their credibility.
Hiking/cutting rates is just one tool in the monetary policy toolbox.
An interesting basic study case is Turkey where the central bank is not raising rates, inflation reached 80%.
So why are central banks raising interest rates to "fight inflation"?
Central Bank's sole job is to fulfil their mandate, CB is not supposed to follow other goals than the ones in the mandate. Two major world's central banks are ECB and Fed.
ECB has sole mandate to keep prices stable which they interpret as average inflation being close to 2% (see ECB). So ECB has to keep inflation at around 2% regardless of what effect it will have on consumers. That is simply inconsequential to ECB.
Fed has dual mandate that requires Fed to (see Fed):
maximum employment, stable prices, and moderate long-term interest rates
The maximum employment is defined as employment at which inflation is not accelerating (see Brookings explainer).
Currently, it is accepted that US is at or very close to maximum/full employment (see WSJ) so Fed managed to fulfill that part of a mandate and only unfulfilled part is to have stable prices.
Most CB's around the world have mandate that either is similar to ECB or similar to Fed.
What do they think they can achieve?
They think they will achieve their job, which is too keep prices stable. Central bank's job is not to pursue some broad measures of well being. Their job is to follow the mandate they have been given.