When the Bank of England buys government bonds, how does it lower the cost of borrowing for other companies?

in simple terms when the bank buys bonds it is increasing the money supply . An increase in the money supply all things equal reduces the interest rate. Now as long as investments are sensitive to changes in interest rates then a reduction in interest rate will cause firms to spend more on machinery, equipment all geared an increasing production. This in itself will also increase GDP through our identity Y=C+I+G+(X-M)