First, to begin with Marx and Engels' concept of historical progress, we have to understand that they viewed history and society as characterized by struggle between social classes determined by their relationship to what they called the "means of production." As they wrote in the Communist Manifesto:
Freeman and slave, patrician and plebeian, lord and serf, guild-master and journeyman, in a word, oppressor and oppressed, stood in constant opposition to one another, carried on an uninterrupted, now hidden, now open fight, a fight that each time ended, either in a revolutionary re-constitution of society at large, or in the common ruin of the contending classes.
As the passage says, each conflict ended in either a restructuring of society through revolution or in the collapse of that society as a whole and the rebirth of a new one. What was different about previous eras was that these conflicts tended to be very complex, involving "gradations" of social rank. Economic changes brought about social changes, like the development of manorial agriculture or guilds in the Middle Ages, that supplanted older means of production.
Marx and Engels argued that the most important development in human history had occurred along with the Industrial Revolution: the development of what he called the bourgeoisie. This was a class of business owners and merchants whose control over the means of production allowed them to profit from industrial development in ways unimaginable in previous centuries. Along with the bourgeoisie developed the proletariat, or working class:
It [modern bourgeois society] has but established new classes, new conditions of oppression, new forms of struggle in place of the old ones. Our epoch, the epoch of the bourgeoisie, possesses, however, this distinctive feature: it has simplified the class antagonisms. Society as a whole is more and more splitting up into two great hostile camps, into two great classes, directly facing each other: Bourgeoisie and Proletariat.
These people labored to produce wealth in factories, but did not receive anything close to the value of their labor because the innovations of the Industrial Revolution allowed them to produce more goods, with more efficiency, than ever before. This development, called the "surplus value of labor" by Marx in later writings, meant that the working class would be "alienated" from the value of the things they labored to produce. The bourgeoisie would always be driven by competition and the profit motive to continually make production more efficient, so this process would only get more and more pronounced with the development of industry. Eventually, Marx and Engels thought, this alienation would become so complete, and the working class would be so large and discontented, that they would rise up to overthrow the bourgeoisie, establishing in its place a classless society. In this way, Marx said, the bourgeoisie had "forged the weapons which bring death to itself" and "called into existence the men who are to wield those weapons—the modern working class—the proletarians." These people, led by a vanguard of Communists that Marx and Engels believed were already seizing control of Europe in the revolutions of 1848, had "nothing to lose but their chains". The authors saw revolution as not only just and desirable, but as inevitable.