After the housing bubble burst many players became indebted (households sat on credits they couldn't pay while the price of their house dropped) or lost wealth (institutional investors as well as ordinary people owned asset-backed securities, i.e. housing credits that were converted into bonds such that they could be traded, which became worthless).
Demand for new houses dropped, banks didn't lend money because they first wanted to reduce their debts / undo their loss of wealth, consumers didn't spend because they had debt issues or feared unemployment, companies didn't invest because the credit markets were frozen. So the problem is that everybody in the private sector is too indebted / doesn't have enough wealth.
Deficit spending: When you reduce taxes in such a situation as many right-wing economists suggested all that happens is that players use the extra money to fix their debts, that's why the government has to directly create demand via e.g. building a road. That was one problem of Obama's stimulus act, too much focus upon tax reductions, too little focus upon directly buying stuff. Note that what happens is that the government pays for the stuff it buys via running into debt. As government bonds are safe assets and as public players wanna get rid of the crappy, worthless, asset-backed securities, as there is a large demand for safe assets interest rates for government bonds are extremely low.
Monetary policy: As interest rates are close to zero the Fed could not reduce them further so they had to do "quantitative easing", i.e. the central bank directly bought stuff and paid for it via money it can print. It didn't buy random stuff but the above mentioned asset-backed securities, so it basically exchanged the fairly worthless bonds nobody wanted with the safe asset money. Like government bonds money is the safe asset which is in high demand.
You can best see why fixing balance sheets matters when you think about why inflation remained low although money tripled after these actions of the Fed. Banks who got this money basically sat on it, they first wanted to fix their balance sheet issues before they lend it out. If you want an analogy, the bucket is nearly empty so you first have to fill it up again before you can use it to pour out water.
Bank bailouts: Obviously they were badly done and a better solution would have been what Sweden did in the nineties during a banking crisis, they didn't just give them "money for nothing" but temporarily socialized the banks. Our problem was that bank managers paid themselves extremely well (of course you could try to empower shareholders such that they can better control managers but this is a corporate governance issue about which I know nothing) after the crisis so to use the water bucket analogy again, filling up the bucket was a good idea but we forgot to fix the hole at the bottom.
Apart from this lemon socialism problem, i.e. that we merely socialized losses, the motivation behind the bank bailout was sound and comparable to quantitative easing: once again you wanna pour large amount of liquidity into the system in order to enable banks to first fix their balance sheets such that they will start to lend again. As the government had to indebt itself to pay for the bailout it is once against a shift of private debt / wealth issues of private players into public debt.
Once you do all this, once you helped everybody to fix their balance sheets banks will lend again, companies will use these loans to invest again and debt-free consumers who do not have to fear unemployment will consume again (helping people with massive debts is not merely a matter of plain decency but also a matter of aggregate demand management, i.e. even if you do not give a damn about these people you want to help them for the sake of ending the recession). Of course public debt and the crappy assets that the Fed owns are a problem ... but unlike private players these public players can deal with it AFTER the recession. As Keynes said, "The boom, not the slump, is the right time for austerity at the Treasury.", i.e. when GDP is growing above average it is time to increase taxes and use the resulting public budget surpluses to reduce public debt. The Fed has to increase interest rates and in general be careful about potential inflation (remember that it bought worthless crap with its own money which implies a long-run risk for money becoming worthless).
The two main ideas are that wealth effects / indebtedness or however you wanna call it matter during a recession (Irving Fisher
already knew it in the thirties and recently Richard Koo
, the guy who basically invented the term balance sheet recession, reminded us that debt matters), that you can exchange risky private debt for safe public debt and that the government can act countercyclically, pull debt out of the hands of private players and postpone serving this debt into the next boom while private players, left on their own, are the victims of a vicious cycle.
This isn't socialism or a progressive love of big government, it is rather the realization that the government is the only force that can end a balance-sheet recession like the current one. Please also note that all these measures are TEMPORARY ones, the government temporarily indebts itself, the Fed temporarily blows up its balance sheet massively. None of this has anything to do with the usual boring political discussion about the LONG-RUN AVERAGE size of government. You can very well be a proponent of small government and demand management. Keynes wasn't a socialist, he made money on the stock market and aptly called his General Theory "mildly conservative".
Sorry for the long post, I hope that it is somehow understandable.