How we cite our quotes: (Section.Paragraph)
Quote #4
The communists are distinguished from the other working-class parties by this only: 1. In the national struggles of the proletarians of the different countries, they point out and bring to the front the common interests of the entire proletariat, independently of all nationality. 2. In the various stages of development which the struggle of the working class against the bourgeoisie has to pass through, they always and everywhere represent the interests of the movement as a whole. (Section2.5)
Working-class political parties usually tend to represent the interests of some workers against other workers. For example, a party representing poor workers might come to power by taking advantage of hatred against immigrant workers. Communists reject this because they believe all workers must unite against the bourgeoisie in order to bring about an end to capitalism. The bourgeoisie that operates around the globe (Section1.19) has global impact; so, Marx believes, should the workers.
Quote #5
The distinguishing feature of Communism is not the abolition of property generally, but the abolition of bourgeois property. But modern bourgeois private property is the final and most complete expression of the system of producing and appropriating products, that is based on class antagonisms, on the exploitation of the many by the few. (Section2.12)
Many fear communism because they believe they will no longer get to have possessions—"Do I still get to have my own toothbrush under communism, or do I have to share it with three other people?"—but Marx's point here is that the bourgeoisie's property is what needs to be abolished, because it depends on previous exploitation of workers. So it's sort of like this: you can keep your handmade toothbrush, but you can't keep your Ferrari.
Quote #6
But does wage-labour create any property for the labourer? Not a bit. It creates capital, i.e., that kind of property which exploits wage-labour, and which cannot increase except upon condition of begetting a new supply of wage-labour for fresh exploitation. Property, in its present form, is based on the antagonism of capital and wage-labour. (Section2.17)
Marx is saying that the proletarian will never be able to earn enough from his or her wage labor to amass property—he or she can barely earn enough to make ends meet. The only way to gain property is to be able to hire other workers and exploit the value they add by their labor in order to create capital.