The financial institutions selling those assets (e.g banks) can then boost the economy by lending this new money to businesses.
Does this work?
Economists generally agree QE can work temporarily when all else fails. But the major problem is that it creates inflation - because there is more money chasing the same amount of goods. And the economic boost will not last, it is a 'sugar high'. Long term the result can be the worst of all worlds: stagflation (rising prices in a contracting economy).
The group that suffers most from QE is those with with savings (because in real terms) their money loses value. The Governor of The Bank of England, Mervyn King, has explicitly conceded this but feels that a second round of QE is essential to avoiding a 'depression' on the scale of the 1930s
QE was eventually used in the early 1990s to counteract deflation in Japan. At best the results were mixed.