I understand that the federal reserve manipulates the federal fund rate in order to change interest rates in the wider banking sector. What I am struggling to understand is how the changing the federal funds rate changes the interest rate charged by commercial banks, e.g. Bank of America, Wells Fargo, etc.
The federal fund rate is a cost for the commercial banks. They borrow at the federal fund rate and then they lend to the consumers. The federal fund rate variation impacts the lending cost of the banks directly. It's how it changes the interest rate charged by commercial banks.
Commercial banks do a lot of lending and borrowing among each other and that is a sign of healthy flow of money in economy. This is because inter-bank activities ensure that enough credit is available and banks are engaging in lending to businesses, individuals etc.
However, sometimes such lending is not what the central bank thinks it should be. Commercial banks may be cautious in lending to public or even to each other. This can happen when there are not enough investment opportunities according to them or because of low market sentiments. Banks at such times play safe and would rather park their money in their central bank accounts in the form of excess reserves.
In such situations, when central bank reduces its rates there is less incentive for banks to park their money with it as their could be some other opportunities which could be more profitable. Secondly, the banks would have cheaper credit which can be used to finance their assets, to give loans and in case they face shortage of cash, central bank window is there to help them. Thus, reduction of rates by central bank can have many transmission mechanisms.
However, there is absolutely no guarantee that such transmissions will happen. This depends on how the commercial banks receive this. How market reacts to this. Transmissions may take months or even years to complete. Within this period, the economy itself might grow or slow down and hence, the actual effect of these rate cuts or hikes could be just opposite to what the central bank intended. Hence, it's a tricky, inexact science and central banks have to work in a lot of uncertainty.