Bond rates move inversely to price.
As the fed buys bonds, it raises the price which lowers the rate.
As rates are lowered for things like mortgages and corporate bonds, people and corporations have more money to spend. Which they do generally spend which stimulates the economy.
Lower rates also cause corporations and people to borrow more which in a fractional reserve banking system actually creates money out of thin air. The reason why corporations borrow more is because with a lower WACC (weight average cost capital) they can invest in more projects (I.e. spend money) for any initiative that has a positive NPV.