Businesses downsize the workforce either (1) because they are in difficulty already, (2) or because they expect a contraction in their business and want to be better prepared, (3) or because they now doubt that their earlier bold plans will work out.
Raising interest rates is often done too late, too much, and too long - which then starts an economic contraction. A "soft landing" is rare so some businesses take measures in advance of that contraction.
But normally interest rate increases are because the economy is expanding too fast (which causes price inflation) - normally that's a low unemployement time. Business response will raise unemployment a little but not much. What WOULD raise unemployement significantly is a contraction because interest rates stayed high too long. An oil shock is a different kind of beast: you get price inflation even though there wasn't excess growth.
Still... 42% is a signal things are out of control. At that stage are businesses still borrowing at all? So will this increase have much effect?
* https://en.wikipedia.org/wiki/Stagflation
It was not 'supposed' to happen in (original/early) Keynesianism, but it occurred in the 1970s in the US, which led some folks to declare Keynesism dead/invalid. People thought about it and analyzed it some, and by the end of the 1970s models were updated with a better understanding of the mechanisms, and every text since about 1980 goes over the details.
It does this by raising the interest rate charged on newly originated loans (why loan at less than the central rate you could get for depositing?), which results in fewer loans (low-return opportunities no longer feasible to finance), which decreases the effective money multiple (due to fractional reserves, availableToLoan:deposited ratio).
On the one hand, this supports the value per currency unit, as less currency chasing same number of goods.
On the other hand, it can decrease net economic activity and blunt its own intent, as you note.
At the end of the day though... it's extremely difficult to run a functioning economy with persistently high inflation. See also: Argentina. So, paraphrasing the new Argentinian president's words, 'This needs measures that will make things worse before it gets better.'
Traditionally, there is the Phillips Curve which states that unemployment and inflation have inverse relationship. https://en.wikipedia.org/wiki/Phillips_curve
So, conversely, you can combat inflation by raising interest rates.
But that's only one cause of inflation. If your economy is unproductive, or if the costs of your supplies go up, then prices will go up regardless of what your interest rates are.
That was what happened to the US (and elsewhere) in the 1970s. Oil prices skyrocketed, and that raises the price of literally everything. Fewer goods were purchased, so companies reduced staff, raising unemployment. Unemployment + inflation is called the "misery index", and it's not easily fixed just with monetary measures.
The US solved that in part by government debt. The government is the "borrower of last resort". It can pump money into the economy even when interest rates are high. That will eventually cause inflation but can also break the cycle of price rises and firings. People buy stuff despite high prices, more workers are hired, and eventually the economy reaches a new equilibrium despite the higher price of supplies.
Raising interest rates is useful to lower inflation when inflation happens on the demand side of things no? By making money more "expensive" it reduces the money in general circulation.
Does it also work when the inflation comes from the supply side of things? Like, a war or natural tragedy that causes a certain commodity to spike in price due to low availability?
The little understood con of interest rate based management of the economy is how much the economy adapts to the interest rate cycle. There are businesses that are less sensitive to interest rates - and there are businesses which can leverage parts of the cycle to benefit from policy depressed rates. It's not necessarily a good thing for the economy to be structured around a policy tool.
Note: Just an interested observer of economics - not a trained economist.
At a fundamental level, the target is consumption and the aim is to reduce it. Inflation is when there’s more money but lesser availability of goods and services that people want. So you either have to increase the production of (or import of) goods and services or reduce the amount of money in circulation. Central banks can only do the latter. The former is the responsibility of the executive and legislature to do with policy measures (short term, medium term, long term) on local and well as the international trade fronts.
Coming back to unemployment, it will likely rise. It will have a spiraling effect on other things (people going broke, loans being defaulted, etc.).
If you’ve read about all the developed countries targeting high inflation, and in particular what the U.S. Fed has been doing, there’s been a lot of talk about a “smooth landing”, which implies that the interest rate hikes will gradually bring down inflation without causing a big decline in GDP growth and related effects. This is mostly a fantasy because in reality nobody has figured out how to get other people, i.e., an entire population, to do something in unison.
While interest rates control ease of access to money, central banks can control the amount of money available through various mechanisms (liquidity control in banks, reserve requirements, etc.). Interest rate increases require time to show up in their effects (several months or quarters). Other measures can have a quicker impact. Businesses, especially larger ones, will have access to bind their effective interest rate to lower values for longer. So these changes won’t show up uniformly everywhere.
All said and done, central banks are typically (IMO) powerless in handling inflation or deflation. Without other policy decisions and policy actions (local and international) by the government, there’s not a lot that central banks can do without inflicting too much damage and getting into a whiplash reaction when things spike out of control.
Longer answer: what you are interested in is called the “transmission mechanism”, and the reduced workforce (and also investment by businesses) is only a part of it. The main part of the transmission mechanism is consumption: when interest rates go up, saving becomes more attractive and people shop less, preferring to spend later. They also face lower asset prices (did you notice how all bonds lost value over the past year? that's what it is) and so choose to spend less.
The inflation-unemployment trade-off that you mentioned is also a part of it. All economists know that there are trade-offs between policy goals, and you have to make judgement calls about what the society's priorities are. If only the voters could vote on “2% lower inflation compared to now” vs “1 percent lower unemployment compared to now”. For better or for worse, they don't do that.
Interest rates play little to no role in this specifically, and actually they are a lot more complicated because policy interest rates are set according to a liquidity effect (more money = lower rates) but natural or market rates move according to the Fisher effect (more money = more nominal spending in the future, thus higher interest rates!). Setting policy interest rates for any length of time thus always involves trying to achieve stability in an inerently unstable fashion, akin to balancing an inverted pendulum. It's not worth worrying too much about those.