If your question is "in what circumstance", 123's comment is correct: when the zero lower bound (ZLB) on the central bank benchmark interest rate is binding, i.e. when that interest rate is zero, and negative rates are undesirable. Sometimes the ZLB can actually be negative, cause banks will still hold some reserves with the central bank under a negative interest rate, but there will still be a lower bound on that. At one point, banks will not keep any deposits at the central bank anymore, and it will be incapable of further influencing the market interest rates via its benchmark rate.
If your question is "how do central banks move inflation back to its target with QE", the (simplified) answer is as follows. There are multiple channels through which QE affects the real economy.
First, the portfolio rebalancing channel denotes the dynamic whereby the large-scale purchase of long-term bonds and other safe assets with newly printed money lowers borrowing costs, raises asset prices and thereby ultimately generates wealth effects and stimulates spending. It does so by compressing the term spread and as a result driving investors out of such assets into riskier, shorter-term ones (Joyce et. al., 2013, p.673-9). This broad channel can be divided into more detailed ones that describe the substitutability of assets in terms of types of risk, safety and liquidity (Krishnamurty and Vissings-Jorgensen, 2011, p.219-223). Altavilla et. al. (2015) find evidence that term spreads have been strongly reduced as a result of ECB QE, mostly by reducing duration and credit risk, and that spill-over into non-targeted assets has been substantial because of the low financial stress at the time of the program’s implementation.
Secondly, via the signalling channel, the announcement of LSAPs increases inflation expectations and lowers long-term interest rates by altering the bank’s incentive structure, exposing its balance sheet to rate rise losses, and thus credibly committing the central bank to keeping interest rates low in the future (Bhattari et. al., 2015). This channel also affects the exchange rate, which depreciates as a result of expectation changes (Andrade et. al., 2016, p. 27-9; Carlo et. al., 2015, p.39).
Thirdly, the liquidity or bank funding channel, which stimulates credits and loans at lower rates by increasing liquidity in the financial sector and depressing money market rates (Bernoth et. al., 2015, p.32).
All these channels contribute towards raising inflation, either indirectly through the inflationary effects of an increase in economic activity, or directly through raising inflation expectations.
