Inflation and money supply increases are normal market dynamics too.
The connection is that aggregate supply and aggregate demand intersect at the current price and output levels of the economy. If aggregate supply contracts, this shifts the output level lower, and the price level higher to compensate. Basically, there is a shortage of basic goods & services needed for the economy to function, so the market price mechanism is trying to incentivize people to work on those sectors that are in shortage.
Drilling in a level, these aggregates are divided into consumption (current needs) + investment (production in expectation of future needs). A supply shock in a basic service like transportation means resources need to be reallocated to cover the increased cost. That means decreased investment, and all of the potential investment opportunities competing for a smaller pool of savings (which is just the excess of current production over current consumption). This drives up the "natural interest rate", the cost of capital in the absence of central bank interventions.
The central bank has a choice between feeling the impact as inflation (where prices rise) or recession (where output drops). If it holds policy rates below the natural interest rate, the money supply increases, and you get inflation. The economy remains out of balance; the market is trying to shift resources from investment to consumption, but if rates are artificially low, investment will be artificially cheap and too many people will be writing software we don't need while we don't have enough truck drivers. If it holds policy rates above the natural interest rate, the money supply shrinks, and you get recession. This is letting the economy come back into equilibrium, but it means that all those software engineers will be laid off and have to get jobs as truck drivers.
The Fed's choice is one of these. They can influence whether we feel the supply shock as a painful recession where lots of people lose their job and have to retrain, or whether we feel it as higher prices as the market tries to correct. But because a supply shock is a real drop in output, it means worsened standards of living regardless.