To Marx, it meant that the state owned the means of production, and determined what goods and services would be provided and at what price (relative to workers' wages). (It turns out that this is a computationally intractable problem, but Marx had no way of knowing that, since the notion of computational complexity wasn't even formalized until nearly a hundred years later.)
In today's modern economy, socialism is nearly nonexistent (Cuba and North Korea are the main exceptions, and Cuba is inching away from that model); "state capitalism", in which the government buys an ownership interest in private firms in order to provide itself some economic benefit (usually related to employment) is somewhat more common, and seen throughout the world, notably in most of Europe, the Middle East, and China. But many other countries, particularly in the Anglosphere, followed Mrs. Thatcher's example and privatized their inefficient state-owned industrial concerns -- notably for this newsgroup including their old-line PTT monopolies.
Because it's utter twaddle: poorly written, poorly conceived propaganda for a particular now-discredited view of political economy. (It wasn't yet discredited at the time that Rand first published it, and of course there's a small but dedicated cadre who still believe in it today.) Oh, and it's morally bankrupt as well, according to the teachings of most of the world's major religions.
No, they just have a different understanding of economics. (On the "far left", at least, an equally discredited one -- for many of the same reasons, it turns out -- but that's an equally tiny group of people.
-GAWollman