It doesn't, or at least not yet. Increasing the money supply might eventually cause inflation, but it's not a mechanical link and doesn't happen automatically. For example, the money might be paid back, and that decreases the money supply again.
Also, for inflation to happen, people will have to try to buy more stuff than what's available, and merchants will have to raise prices. Although we've seen shortages for a few things, that kind of overall spending is unlikely to happen until the recovery phase when people are feeling confident again.
Other than the staggering amounts, it's not really different from any other bank making a large loan.
There's more to inflation than CPI. For example, the entire stock market is being inflated right now with these dollars. We should be experiencing deflation because many classes of assets should be crashing harder than they are. Keeping things steady when they should be deflating is still taking my money through inflationary forces.
There aren't many people who benefit from a stock market crash, which is why the Fed is propping things up. I happen to have money to invest when the price is low enough, but that's unusual, and complaining about not having that opportunity (yet) would be "world's smallest violin" territory.
An opportunity cost is not a real cost. There are always missed opportunities.
Stocks are not the only thing that would be falling rapidly. Real estate is another. What's happening here is that the Fed is making sure that the people who are rich and powerful stay rich and powerful. Otherwise, why not let average people get functionally infinite loans at zero interest? Why can a bank/corporation sell their depressed assets to the Fed at before crash prices but I can't? Every privilege given to incumbent wealth is a middle finger to the average worker.
That's kind of the point of inflation though, isn't it? If you accept that money exists to facilitate trade and improve on the barter system (not necessarily a reasonable assumption, especially since money is flexible enough that it could very well serve other purposes if we wanted and allowed it to), then it's a bit odd to let it hold its value in perpetuity because the delayed nature of such transactions no longer represents the barter system that money was meant to replace.
As a simplified example, suppose that 1 duck is generally worth 2 chickens and that monetary prices reflect this (e.g. ducks are worth $10 and chickens $5). I sell you a duck for $10. Time goes on, and the relative values swap -- ducks are worth $5, and chickens are worth $10. I buy two ducks from you for $10. Those two _monetary_ transactions if compressed in time into the bartering that money was meant to replace would represent me trading one of my ducks for two of yours, which assuming no other hidden variables is an unrealistic trade that would never happen.
The relative values of goods shift constantly, and arguably a dollar yesterday really isn't conceptually the same thing as a dollar today -- even though they can both be represented with the same slip of paper. Inflation is one strategy to align money more closely with its role as a medium of exchange.
standard disclaimers -- opinions are my own not my employer's, I'm not a lawyer, I'm not an economist, the above is a simplified model and blatantly ignores huge swathes of fiscal policy, ....
Also, for inflation to happen, people will have to try to buy more stuff than what's available, and merchants will have to raise prices. Although we've seen shortages for a few things, that kind of overall spending is unlikely to happen until the recovery phase when people are feeling confident again.
Other than the staggering amounts, it's not really different from any other bank making a large loan.