"The working men have no country." What did Karl Marx mean by his statement, and why was this assertion considered to be subversive and dangerous...

By this statement, Marx and Engels mean that the nation, as traditionally conceived, is a bourgeois nation. This is to say that a nation with a state of its own and a discrete political history has been shaped by the bourgeoisie and, thus, fully reflect their values. The proletariat, therefore, are not a full-fledged part of this nation; it is not their nation to begin with:

The working men have no country. We cannot take from them what they have not got. (Emphasis added).

The nation, as understood by the bourgeoisie, gives the false impression that class distinctions are somehow less important than one's national or ethnic identity. At the same time, it perpetuates class distinctions through the inherently exploitative capitalist system. We see this during times of war, when everyone is urged to put aside their differences of class in order to unite in the face of a common enemy. However, this is simply a more explicit manifestation of a permanent phenomenon. However, the days of the bourgeois nation are numbered, and, paradoxically, it is capitalism that will ultimately be responsible for its demise:

National differences and antagonisms between peoples are vanishing gradually from day to day, owing to the development of the bourgeoisie, to freedom of commerce, to the world market, to uniformity in the mode of production and in the conditions of life corresponding thereto.

Yet, for the foreseeable future, the struggle for supremacy between the bourgeoisie and the proletariat must still take place within the nation:

Though not in substance, yet in form, the struggle of the proletariat with the bourgeoisie is at first a national struggle. The proletariat of each country must, of course, first of all settle matters with its own bourgeoisie.

The revolution needs to be established in each individual nation first before it can spread worldwide. The traditional bourgeois nation-state as it stands is finished; the cynical chauvinism and patriotism used to mask the grotesque inequality and exploitation of the capitalist system will then go soon after. 

This is a highly subversive notion as it constitutes a direct challenge to the existing system. If the proletariat realizes that the nation as it currently exists is not their own, they can also build a nation of their own. The resulting implications are truly revolutionary. The national struggle is a class struggle. Only, the bourgeoisie do not recognize this fact, preferring to peddle the fiction that we are all in it together. The proletariat, however, see things differently. They know that because the nation is founded on the dominance of a single class, they must themselves become the dominant class. However, this in itself is not enough. The national struggle is merely a prelude to a wider, worldwide revolution against capitalist oppression:

In proportion as the exploitation of one individual by another is put an end to, the exploitation of one nation by another will also be put an end to. In proportion as the antagonism between classes within the nation vanishes, the hostility of one nation to another will come to an end.

Capitalism, and the antagonistic system of national and ethnic rivalries it engenders, will one day come to an end. In its place will arise an international proletarian brotherhood, one founded on the solidarity of the oppressed. Initially, the basic political unit will remain that of the nation-state, but this will in due course wither away as worldwide communism is finally established.