Keynes was a brilliant and influencial economist, and many very bright men have analyzed and expanded upon his theories.
But, I'll give you a few points that I find most interesting.
First off, yes, Keynesians do go in for government intervention. Keynes repudiated classical theories that claimed economies naturally gravitated towards full employment, for instance. Classical theories assumed that all available exploitable economic resources would be utilized, since doing so is only rational. Keynes recognized that economies could run below full utilization, and thus, below full employment, and, that while price-levels tend to self-adjust in a free market, resource utilization does not have a strong self-adjustment mechanism.
In his day, the Business Cycle had been undergoing successively more dramatic highs and lows (culminating with the Great Depression). So, understanding and doing something about economic downturns was a major thrust of his work. Which leads to what I find Keynes's most compelling idea: the Keynsian Cross.
It's a graph of two lines, spending & savings, as income increases. At low income, people tend to spend more than they have, engaging in borrowing or dis-savings; at high income, they tend to save more than they make. At one point, the two lines cross, thus the name. The upshot for this in regards to economic depressions is that drop offs in investment (which often happen early in a downturn), if not countered by comparable reductions in savings, leave money sidelined, and further slow the economy, leading to a persistent depression.
The danger Keynesian interventionism seeks to avoid, then, is excessive savings durring downturns. It does this by increasing government spending (funded through borrowing or progressive taxation), particularly on infrastructure, and by lowering interest rates to increase money available to drive demand.
Progressive taxation and government borrowing take money out of the 'excessive savings' pool. Higher tax rates on the 'rich' reduce the income of those on the high end of the Keynesian Cross, pushing them down, and reducing thier savings rates. Borrowing takes money from the 'excessive saving' pool, and spends it, obviously. Such 'fiscal stimulus' on the government's part increases demand. Reducing interests rates induce dis-savings on the part of lower incomes (who are losing less of thier income due to progressive taxation), again, consuming the 'excessive savings' and building demand. Increasing demand, then, creates an environment in which opportunities exist to increase suply (via increased utilization of resources - labor and investment capital), and thus pull out of the recession.
This counter-cyclical intervention was generally accepted, and was used through the 1970s, encouraging ever-falling rates of saving and investment, and, eventually, breaking down in a period of economic contraction, high inflation and unemployment ('stagflation').
Though it was seen as a failure of Keynesian economics, it really made sense in light of it. Decades of progressive taxation, and counter-cyclical econmic stimulus had erroded savings and investment, to the point that, instead of excessive savings, we had inadequate savings, which hamstrung the economy in a different way. Instead of under-utilization being the prime problem, lack of investment had become the problem.
Neo-classicism became influencial in the Regan years, and taxes were made less progressive, and lowered in general, to free up money at the income levels that would invest it. Investment improved, some, and, the country recovered, rather painfully, from the 70s.
Today, we are looking at economic problems that have responded to neither neo-classical 'supply side' stimulus, nor Keynesian counter-cyclical fiscal stimulus (so far). Said fiscal stimulus has been turned up to '11,' though, we should be seing some results - even if only 70s style hyperinflation.
