How Can Marxism Develop the Society which Capitalism Cannot ?

It can't.

Marx did not have a plan. He was an abstract theorist. He had no connection to the real world - he never had a paying job, he never organized anything involving other people. He was just sitting on his butt and daydreaming.

In his dream, he thought that people would just stop looking after their own self-interest from the very moment that they could no longer own means of production, and the world will live in perfect harmony.

FYI In modern economy the primary means of production are human brains, or as we call it in Economics, human capital. So there is no longer a distinction between workers and owners of capital.

It should be obvious.

With capitalism, the rich get richer and the poor get poorer, to the point where the poor have barely enough to survive.

If they band together and use their power cooperatively, then can take a larger percentage of the total product, produce more, and make everyone except the very rich better off.

You see a little of this today. In the U.S. you have people agreeing that increasing income inequality is the natural result of the free market and therefore is not a problem but is good news:
http://reason.com/archives/2009/05/26/nostalgianomics
And while classical economic theory says wages should go up with productivity, in practice, they don't:
http://www.nytimes.com/2006/08/28/business/28wages.html

But the fact of the matter is that when Democrats hold the White House, the economy grows faster than when the Republicans hold the White House AND income inequality declines:
http://rodrik.typepad.com/dani_rodriks_weblog/2008/03/american-politi.html
This says that government policies make a difference and they can make things better for everyone.

Furthermore, increasing income inequality results in lower rates of income mobility, innovation, education, etc. - all the factors that contribute to economic growth
http://www.greenchange.org/article.php?id=5975

If you look at the Scandinavian countries, with much lower income inequality, which some Americans term as "socialist", they've been doing better economically and their workers have been doing better as well.
http://economistsview.typepad.com/economistsview/2010/11/when-is-economic-growth-good-for-the-poor.html
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=897746

And compare the incomes of the poor (bottom decile) in the U.S. with the poor in Europe:
http://economistsview.typepad.com/.a/6a00d83451b33869e20133f44756f7970b-popup
The lowest 10% are poorer in the U.S. than the lowest 10% is in Greece.

So with suitable government policies, a country can have a high standard of living for all its people.

And the evidence is that Americans like this idea:
http://baselinescenario.com/2010/09/28/americans-want-to-live-in-sweden/

Furthermore, pure capitalism means no government regulation. As we have seen, less regulation means more crises. (The current recession was caused by lack of regulation in the financial sector. Just look at all the recession the U.S. had before the Fed was set up to do some regulation.)

This isn't to say that a revolution of the proletariate would bring about this kind of society. But it is clear that such a society can do things which purer capitalism can not.

(Not that the U.S. is pure capitalism. It is a mixed economy, but does give far more power and weight to the capitalists than do economies such as those in Scandinavia.)
http://en.wikipedia.org/wiki/Corporate_capitalism
vs.
http://en.wikipedia.org/wiki/Rhine_Capitalism

Note that neither Russia nor China are good examples of Marx' ideas. Neither of those was an industrialized economy when the Communists took over; both were peasant economies; neither saw a revolution by the workers.