As best I understand it Keynes has two essential principles that differentiate his economic theory from classical economics: Keynes differentiates uncertainty from risk, and he emphasizes demand over supply.
The first principle is (now that Keynes has explained it) pretty obvious. There are future events we know for sure (if I drop a glass, it will certainly fall to the floor and break), and future events we don't know for sure. In the latter category, there are future events for which we know the probability distribution and those we don't. The first is risk: We don't know how the dice will fall, but we know very precisely the probability distribution of the possible outcomes. The second is uncertainty: we have no way of knowing, for example, the actual probability that the dice might be loaded. We might make guesses from the past, but we can't rationally work out the probability distribution from first principles.
This principle changes the character of two key assumptions of classical economics. The first is the Rational Expectations Hypothesis (REH). Leaving aside precisely what constitutes "rationality", REH says that the price of a commodity includes a rational evaluation of the risks inherent in owning that commodity. The price of a house includes the risk that the house will burn down; the price of a mortgage includes the risk that the borrower will default. According to Keynes, because we cannot by definition rationally evaluate uncertainty, the price of a commodity cannot include such a rational evaluation. All the price can do is reflect our subjective feelings about the uncertainty.
Likewise, the Efficient Market Hypothesis (EMH), which says that even though individuals might over- or under-estimate risk, these errors will be randomly distributed around the correct evaluation; in a large enough market, these errors will cancel out and the price of a commodity will reflect its accurate risk. Again, since we cannot rationally evaluate uncertainty, there is no "correct" evaluation for the errors to be randomly distributed around.
Keynes doesn't invalidate REH or EMH; under Keynes these hypotheses say that the price of a commodity reflects what we know about the risks and what we believe about the uncertainty, and the errors of individuals in a market will be distributed around at least the accurately known risks. A market cannot, however, accurately account for uncertainty. Keynes thus proposes a role for government in managing and reacting to fundamental economic uncertainties.
Keynes less obvious principle, a corollary to the principle of uncertainty, is that effective demand cannot be treated even theoretically as frictionless. Under classical economics, money is not itself a commodity: it has zero use-value. A person with money must necessarily immediately either spend it (purchasing something she can consume) or invest it (use it to create more commodities). People should hold only a trivial amount of money in as actual cash.
Keynes, however, holds that money-as-cash is a commodity: it directly represents subjective doubt about uncertainty. When people have little doubt, the don't hold much cash. When people have a lot of doubt (they can't actually know the probability distribution of the uncertainty) they hold a lot of cash.
What the Federal Reserve (and other central banks) typically do is adjust the money supply (or, more precisely, immediate changes to the money supply) to accurately reflect people's trade-offs between consumption and investment. When there aren't enough goods to meet effective demand, inflation rises, the Fed raises interest rates (i.e. it costs more for an individual bank to create money). Since consumption is debt-funded, higher interest rates encourage more investment and less consumption. When there are too many goods, inflation falls, unemployment rises, so the Fed lowers interest rates to increase consumption and decrease investment.
However, when people have a lot of doubt, changing the money supply through interest rates is ineffective. People keep cash itself for its use-value to assuage uncertainty and they neither spend it nor invest it. The Fed cannot raise rates; the underlying problem is the oversupply of goods. If the Fed were lower interest rates below 0%, then the creation of new money is exactly balanced by increased doubt in the value of that money. Monetary policy thus loses traction to affect the economy. Keynes thus proposes fiscal policy: The government, since it does not have to be productive to get money, can act intentionally contrary to doubt. Thus when doubt about uncertainty causes people to hoard cash, the government borrows that cash and spends it. It almost doesn't matter what the government spends the money on: we can of course do better, but as Keynes notes, burying money in the ground and letting people hunt for it would accomplish the desired effect.
Keynes didn't get everything right (nor did Newton, Darwin, Einstein or Heisenberg) but he got enough right, and what he got right is — once explained — obviously right.
While Keynes was at least to some extent a "moral" progressive, his economic theory is not inherently progressive, at least not at first glance. His theory works just as well under "third-world" capitalism (where effective demand is concentrated in a minuscule ruling class and an only slightly smaller middle class) as under "first-world" capitalism (where effective demand is somewhat more distributed to the working class and a larger middle class). "Crony" capitalism is precisely Keynesian economics applied to third-world political conditions.
It's also worth noting that the United States government operated (mostly) under Keynesian principles for almost thirty years. (Even Richard Nixon famously said, "We are all Keynesians now.") Under Keynesian economics, the capitalist ruling class did pretty damn well: they pretty much conquered the entire world.
It's also worth noting that high unemployment is not (the gyrations of some anti-capitalist academics notwithstanding) in the mutual interest of the capitalist ruling class, nor is it in any one member's individual interest. Even in the worst third-world conditions, unemployed labor is labor that's not being exploited; unemployment basically leaves money lying on the ground.
So we have to ask ourselves a few questions. Why do we actually have high unemployment right now in the United States? Why is this unemployment expected to persist for many years? And, most importantly: why did the Randian faction of the capitalist ruling class wage a bitter and desperate war for decades against Keynesian economics?