Close. The federal reserve (or any central bank) will lend money to other banks at a rate just below the target rate, and it will borrow from other banks at a rate just above. Because banks can borrow and lend largely risk free at those two rates, banks will transact amongst themselves at a rate in between. This is how the federal reserve makes banks transact at the target rate.
If you're asking why banks lend money to each other, it's mostly because banks need liquidity (e.g. $ in an account) in order to do transactions and meet legally mandated reserve requirements. If I'm a bank, rather than constantly be sitting on a huge pile of money for the (relatively) rare event that I have a huge transactions to make, I'll mostly do something more useful with my money, and then borrow it from another bank on a short-term basis when I need a bunch of cash.
The fed does this for banks that can't find another bank to transact with, albeit at a slightly higher rate. The fed will also borrow money from banks at a slightly lower rate. This causes the banks to naturally lend among each other between those two rates. This is how the fed controls the interest rate.
The other thing the fed does is buy treasuries. I think this is where the "printing" of money happens. The fed buys a treasury on the market and pays for it by printing money and placing it in the appropriate account. This is how the fed controls money supply.
Yes, but the main effect is how much it costs to “buy” money - if the fed rate is zero then companies can often get money at 1% or so - which means if they have a way of making only 1.5% on the money it’s worth doing it and they grow.
When the rates rise, it’s no longer worth doing these marginal businesses and so growth slows down. You’re not going to borrow at 5% to make 3%.
