How the base rate works
If the Bank of England wants to increase the expansion of the money supply, it will reduce the base rate to incentivise commercial bank lending. Lending creates customer deposits increasing the money supply, and as customers (people and businesses) move their deposits to other banks, reserves are run down at the bank that created them (and run up at other banks).
The bank benefits by getting a higher rate of return on the loans it created than on the interest paid by the Bank of England, albeit by taking on higher risk.
If the Bank of England wants to reduce the expansion of the money supply, it will increase the base rate to reduce commercial bank lending and increase reserves.
Commercial bank balancing act
As banks increase and decrease lending, reserves flow back and forth between banks, so a bank has to adjust the interest it pays on deposits and savings to control the flows because deposits will chase the highest interest rates. A bank that has increased its lending may need to increase its reserves to keep the Reserve Ratio from dropping too low, so the bank would have to increase the interest it paid on deposits and savings accounts.
This is how the Bank of England base rate heavily influences the borrowing and savings rates of commercial banks. Mortgage rates are tied to this rate and go up and down with it, as are business loan rates. Similarly, interest rates paid on deposits and savings accounts do the same.