[roussanoff]

Short answer: it's complicated.

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.

[zozbot234]

The transmission mechanism is not actually via raising interest rates in any real sense, it's by draining money from the economy - selling assets for money, and then retiring that money. This leaves economic actors with lower money balances than they wish to hold at the given level of nominal spending (price level × real activity) which in turn incents them to spend less in all kinds of ways. Which then leads to lower expectations of the future price level (as well as future real activity), which works to reel in the inflationary spiral.

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.